Homological algebra in bivariant K-theory and other triangulated categories. II
aa r X i v : . [ m a t h . K T ] D ec HOMOLOGICAL ALGEBRA IN BIVARIANT K-THEORY ANDOTHER TRIANGULATED CATEGORIES. II
RALF MEYER
Abstract.
We use homological ideals in triangulated categories to get a suffi-cient criterion for a pair of subcategories in a triangulated category to be com-plementary. We apply this criterion to construct the Baum–Connes assemblymap for locally compact groups and torsion-free discrete quantum groups. Ourmethods are related to the abstract version of the Adams spectral sequence byBrinkmann and Christensen. Introduction
The framework of triangulated categories is ideal to extend basic constructionsfrom homotopy theory to categories of C ∗ -algebras. It provides a uniform setting forvarious problems in non-commutative topology, including homotopy colimits andMayer–Vietoris sequences, universal coefficient theorems, and generalisations of theBaum–Connes assembly map (see [16–20]). More specifically, the Baum–Connesassembly map for coactions of certain compact Lie groups, which is studied in [17],is always an isomorphism and it is closely related to a universal coefficient theoremfor equivariant Kasparov theory by Jonathan Rosenberg and Claude Schochet ([22]).Universal coefficient theorems for Kirchberg’s bivariant K-theory for C ∗ -algebrasover certain finite topological spaces are derived in [19, 20].This article continues [18], which deals with a framework for carrying over famil-iar notions from homological algebra to general triangulated categories. Before weexplain what this article is about, we outline some important ideas from [18].The localisation (or total derived functor) of an additive functor between Abeliancategories is a functor between their derived categories. Mapping chain complexesto chain complexes, it belongs to the world of triangulated categories by defini-tion. Although the more classical derived functors originally live in the underlyingAbelian categories, they can be carried over to triangulated categories as well.Both localisations and derived functors require additional structure on a trian-gulated category to be defined. For the localisation of a functor, we specify the subcategory to localise at, consisting of all objects on which the localisation van-ishes. For its derived functors , we specify an ideal , consisting of all morphisms onwhich the derived functors vanish.The idea to use ideals in triangulated categories goes back to Daniel Chris-tensen [7]. Some important related concepts are due to Apostolos Beligiannis [3],who uses a slightly different but equivalent setup, which is inspired by the notionof an exact category in homological algebra.A homological ideal in a triangulated category T is, by definition, the kernel of astable homological functor (see [18]). Such an ideal I allows us to carry over variousnotions of homological algebra to T . The ultimate explanation for this is that ahomological ideal generates a canonical homological functor to a certain Abeliancategory, namely, the universal I -exact stable homological functor H I : T → A I T .All homological notions in T defined using the ideal I reflect familiar notions in Mathematics Subject Classification. this Abelian category. The homological algebra in the target Abelian category A I T provides a rough Abelian approximation to the category T .An interesting and typical example is the G -equivariant Kasparov category KK G for a countable discrete group G . Let I be the ideal defined by the K-theory functor,that is, an element of KK G ( A, B ) belongs to the ideal if it induces the zero mapK ∗ ( A ) → K ∗ ( B ) . The resulting Abelian approximation A I ( KK G ) is the categoryof all Z / -graded countable modules over the group ring Z [ G ] , and the universalfunctor maps a C ∗ -algebra A with an action of G to its K-theory, equipped withthe induced action of G (this is a special case of a result in [18]).Notice that the passage to the universal functor adds the group action on K ∗ ( A ) .Forgetting this group action does not change the ideal defined by the functor, but itkills most interesting homological algebra. (In Section 7, we will actually considera smaller ideal in KK G that is more closely related to the Baum–Connes conjecture,but leads to a more complicated Abelian approximation.)The Abelian category A I T is usually not a localisation of T : we must modifyboth morphisms and objects to get an Abelian category. Instead, it is describedin [3] as a localisation of the Abelian category containing T constructed by PeterFreyd. The main innovation in [18] is a concrete criterion for a stable homologicalfunctor to be universal, which involves its partially defined left adjoint. Using thiscriterion, we can often find rather concrete models for the universal functor – as inthe example mentioned above – and then compute derived functors associated tothe ideal.What do the derived functors of a homological functor on T tell us about theoriginal functor? In general, these derived functors are always related to the originalfunctor by a spectral sequence, whose convergence we will discuss below. Thisresult is mainly of theoretical importance because spectral sequence computationsare almost impossible without additional simplifying assumptions. But given howmuch information is lost by passing to an Abelian category, we cannot hope formuch more than a spectral sequence.The spectral sequence that links a functor to its derived functors was alreadydiscovered in the 1960s before triangulated categories became popular. First FrankAdams treated an important special case in stable homotopy theory – the Adamsspectral sequence [1]. This was reformulated in an abstract setting by Hans-BerndtBrinkmann [6]. Daniel Christensen [7] formulated the Adams spectral sequence inthe setting of triangulated categories, apparently unaware of Brinkmann’s work.Given the sources of the spectral sequence, we call it the ABC spectral sequence here. We describe its construction and its higher pages in greater detail thanprevious authors and weaken the assumptions needed to guarantee its convergence.I was drawn towards this theory because similar ideas provide an effective methodto prove that pairs of subcategories are complementary; this is the most difficulttechnical aspect of the construction of the Baum–Connes assembly map in [16]. InSection 7, we first apply our new criterion to the group case already treated in [16]and then define an analogue of the Baum–Connes assembly map for all “torsion-free” discrete quantum groups. More precisely, we construct an assembly map forall discrete quantum groups, but since this map does not take into account torsion,it is not the right analogue of the Baum–Connes assembly map unless the quantumgroup in question is torsion-free.A built-in feature of our new assembly map is that its domain is computed bya spectral sequence – the ABC spectral sequence – whose second page is quiteaccessible. The spectral sequence computation is very difficult, but an operatoralgebraist might consider it to be a topological problem, that is, Someone Else’sProblem. His own problem is to find out when the assembly map is an isomorphism.
OMOLOGICAL ALGEBRA IN BIVARIANT K-THEORY 3
Given our experience with the group case, this should happen often but not always.So far – besides classical groups – only the duals of certain compact Lie groupsand quantum SU (2) have been treated in [17] and in [24], respectively. For thealternative approach by Aderemi Kuku and Debashish Goswami in [11], it is unclearwhether the domain of the assembly map is computable by topological methods.Our criterion for complementarity of two subcategories is also useful in situationsthat have nothing to do with bivariant K-theory. The improvement upon similarcriteria in [3] is that we can cover categories that are not compactly generated:what we need is an ideal with enough projective objects that is compatible withcountable direct sums. This assumption is still satisfied for the ideals that appearin connection with the Baum–Connes assembly map, although the categories inquestion are probably not compactly generated.More precisely, the criterion is the following. Let I be a homological ideal in atriangulated category T . We assume that T has countable direct sums and that theideal I is compatible with countable direct sums in a suitable sense. Furthermore,we assume that I has enough projective objects. Let P I ⊆ T be the class of I -projective objects in T and let h P I i be the localising subcategory generated byit, that is, the smallest triangulated subcategory that is closed under countabledirect sums and contains P I . Finally, let N I be the subcategory of I -contractibleobjects. Under the assumptions above, the pair of subcategories ( h P I i , N I ) iscomplementary, that is, T ∗ ( P, N ) = 0 whenever P ∈∈ h P I i and N ∈∈ N I , and anyobject A ∈∈ T is part of an exact triangle P → A → N → P [1] with P ∈∈ h P I i and N ∈∈ N I . Equivalently, the subcategory N I is reflective , that is, the embedding N I → T has a right adjoint functor.Our proof also provides the following structural information on the category h P I i . First, we get an increasing chain ( P n I ) n ∈ N of subcategories, consisting of theprojective objects for the ideals I n ; these can also be generated iteratively from P I using exact triangles. We show that any object of h P I i is a homotopy colimit ofan inductive system P n with P n ∈∈ P n I . Notation 1.1.
We write f ∈ C for a morphism and A ∈∈ C for an object of acategory C . We denote the category of Abelian groups by Ab . We usually write T for triangulated, A for Abelian, and C for additive categories. The translationautomorphism in a triangulated category is denoted by A A [1] .2. Homological ideals, powers, and filtrations
The convergence of a spectral sequence always involves a filtration on the limitgroup. Hence we expect a homological ideal I in T to generate filtrations on the cat-egory T itself and on homological and cohomological functors on T . After recallingsome basic notions, we introduce these filtrations here.We will use the results and the notation of [18]. In particular, a stable categoryis a category with a translation automorphism, denoted A A [1] , and a stable functor is a functor F together with natural isomorphisms F ( A [1]) ∼ = ( F A )[1] forall objects A .Let F : T → A be a stable homological functor from a triangulated category T to a stable Abelian category A . We define an ideal ker F in T by ker F ( A, B ) := { ϕ ∈ T ( A, B ) | F ( ϕ ) = 0 } . Ideals of this form are called homological ideals . A homological ideal is used in [18] tocarry over various notions from Abelian to triangulated categories. This includes I -epimorphisms, I -exact chain complexes, I -exact functors, I -projective objects,and I -projective resolutions. The first three of these can be tested using the func-tor F ; for instance, a chain complex with entries in T is I -exact if and only if its RALF MEYER F -image is an exact chain complex in the Abelian category A . Projective objectscan only be described in terms of F if F is the universal I -exact stable homologicalfunctor, see [18]. We also call a morphism an I -phantom map if it belongs to I .Most of our constructions require T to contain enough I -projective objects – thatis, any object should be the range of an I -epimorphism with I -projective domain.This is equivalent to the existence of I -projective resolutions for all objects. Remark . Daniel Christensen uses a somewhat different terminology in [7]. His projective classes ( I , P ) turn out to be the same as a homological ideal I withenough projective objects together with its class P = P I of projective objects.The ideal I in a projective class is homological because, in the presence of enoughprojective objects, the universal homological functor with kernel I is well-defined.There are two ways to construct this universal functor, which involve a localisationof categories in one step. Apostolos Beligiannis [3] first embeds the category T intoan Abelian category and then localises the latter at a Serre subcategory. The au-thors use the heart of a t-structure on a suitable derived category of chain complexesover T in [18, §3.2.1]. In both cases, the morphisms in the relevant localisationscan be computed using projective resolutions, so that the localisation is again acategory with morphism sets instead of morphism classes.2.1. Powers and intersections of ideals.
At first, we do not care whether theideals we are dealing with are homological. Let C be an additive category. If ( I α ) α ∈ S is a set of ideals, then the intersection T I α is again an ideal. If I , I ⊆ C are ideals, define I ◦ I ( A, B ) := { f ◦ f | f ∈ I ( X, B ) , f ∈ I ( A, X ) for some X ∈∈ C } . This is a subgroup of C ( A, B ) because we can decompose f ◦ f + f ′ ◦ f ′ as A (cid:16) f f ′ (cid:17) / / X ⊕ X ′ ( f f ′ ) / / B. Thus I ◦ I is an ideal in C . We have I ◦ I ⊆ I ∩ I .Now the powers of an ideal I ⊆ C are defined recursively: we let I := C consistof all morphisms and define I n := I n − ◦ I for ≤ n < ∞ . The sequence of ideals ( I n ) n ∈ N is decreasing, and we have I m ◦ I n = I m + n for all m, n ∈ N . If I n = I n +1 for some n ∈ N , then I n = I N for all N ≥ n .We also let I ∞ := T n ∈ N I n and I n ∞ := ( I ∞ ) n . In general, the ideals I ◦ I ∞ and I ∞ ◦ I may differ from I ∞ (see the remark after Proposition 4.7). Theorem 3.27shows that I n ∞ = I ∞ for all n ≥ if I is compatible with countable direct sums.Now we replace the additive category C by a triangulated category T and restrictattention to homological ideals. It is not obvious whether the powers of a homolog-ical ideal are again homological. If I = ker F , then a functor with kernel I ⊂ I contains more information than F because it has a smaller kernel. Therefore, wecannot hope to construct such a functor out of F .Nevertheless, I expect that products and intersections of homological ideals areagain homological, at least if the categories in question are small to rule out settheoretic difficulties with localisation of categories. A proof could use Beligiannis’axiomatic characterisation of homological ideals. Since we only need the mucheasier case where there are enough projective objects, I have not completed theargument. Proposition 2.5 and Theorem 3.1 in [3] show that our “homologicalideals” are exactly the “saturated Σ -stable ideals” in Beligiannis’ notation. Clearly,products and intersections of Σ -stable ideals remain Σ -stable, and intersections ofsaturated ideals remain saturated. It is less clear whether products of saturatedideals remain saturated; the proof should involve the octahedral axiom. OMOLOGICAL ALGEBRA IN BIVARIANT K-THEORY 5
Here we only consider the easy case of ideals with enough projective objects,where we can describe which objects are projective for products and intersections:
Proposition 2.2 ([7, Proposition 3.3]) . Let I and I be homological ideals in T with enough projective objects. Then I ◦ I is a homological ideal with enoughprojective objects. An object A of T is I ◦ I -projective if and only if there are I j -projective objects P j and an exact triangle P → P → P → P [1] , such that A is a direct summand of P . Proposition 2.3 (see [7, Proposition 3.1]) . Let ( I α ) α ∈ S be a set of homologicalideals in T with enough projective objects. Suppose that T has direct sums of car-dinality | S | . Then I S := T α ∈ S I α is a homological ideal with enough projectiveobjects. An object A of T is I S -projective if and only if there are I α -projectiveobjects P α such that A is a direct summand of L α ∈ S P α . We may use Christensen’s results because of Remark 2.1.
Definition 2.4.
Let I be a homological ideal in a triangulated category T withenough projective objects. We write P I for the class of I -projective objects, and P n I for the class of I n -projective objects for n ∈ N ∪ {∞} .The class P I is always closed under direct summands, suspensions, and directsums that exist in T .Propositions 2.2 and 2.3 show that the powers I n for n ∈ N ∪ {∞} have enoughprojective objects. Moreover, P n I for n ∈ N consists of all direct summands ofobjects A n ∈∈ T for which there is an exact triangle A n − → A n → A → A n − [1] with A n − ∈∈ P n − I , A ∈∈ P I ; and P ∞ I consists of all retracts of objects of theform L n ∈ N A n with A n ∈∈ P n I . The phantom castle introduced in Definition 3.15explicitly decomposes objects of P n I into objects of P I ; essentially, its constructionis the proof of Proposition 2.2. Example . Let I be a homological ideal with enough projective objects. If I = I , then Proposition 2.2 implies that P I is closed under extensions. Since thissubcategory is always closed under direct summands, suspensions, isomorphism,and direct sums, P I is a localising subcategory of T .Conversely, if P I is a triangulated subcategory, then I and I have the sameprojective objects. Since an ideal with enough projective objects is determined byits class of projective objects, this implies I = I .Homological ideals with I = I , but possibly without enough projective objects,play an important role in [14] as a substitute for localising subcategories.We usually know very little about the Abelian approximations generated by I n for n ≥ , even if the situation for I itself is rather simple. Derived functors for I and I do not seem closely related. This is particularly obvious in cases where I = 0 and I = 0 . For instance, this happens if I is the kernel of the homology functor onthe derived category of the category of Abelian groups. Here the universal I -exactfunctor is the homology functor to Ab Z ; the universal I -exact functor is the Freydembedding of the derived category into an Abelian category.2.2. The phantom filtrations.
Let I be an ideal in an additive category C . Since I α ⊆ I β for α ≥ β , we get a decreasing filtration C ( A, B ) = I ( A, B ) ⊇ I ( A, B ) ⊇ I ( A, B ) ⊇ · · · ⊇ I ∞ ( A, B ) ⊇ { } , called the phantom filtration [3]. We shall also need related filtrations on contravari-ant and covariant functors on C . RALF MEYER
Let G : C op → Ab be a contravariant functor and A ∈∈ C . We define a decreasingfiltration G ( A ) = I G ( A ) ⊇ I G ( A ) ⊇ I G ( A ) ⊇ · · · ⊇ I ∞ G ( A ) ⊇ { } on G ( A ) by I α G ( A ) := { f ∗ ( ξ ) | f ∈ I α ( A, B ) , ξ ∈ G ( B ) for some B ∈∈ C } . If we apply this construction to the representable functor C ( , B ) we get back thefiltration I α ( A, B ) on C ( A, B ) . If G is compatible with direct sums, then (5.3)asserts that T n ∈ N I n G ( A ) = I ∞ G ( A ) .The functoriality of G restricts to maps I β ( A, B ) ⊗ I α G ( B ) → I α + β G ( A ) , f ⊗ x f ∗ ( x ) , for all α, β . In particular, I α G is a contravariant functor on C . The ideal I β acts trivially on the subquotients I α G ( A ) / I α + β G ( A ) , which therefore descend tofunctors on the quotient category C / I β .We may also view G as a right module over the category C and C / I α as a C -bimodule. The quotient G/ I α G corresponds to the right C -module G ⊗ C C / I α .For a covariant functor F : C → Ab , we define an increasing filtration { } = F : I ( A ) ⊆ F : I ( A ) ⊆ F : I ( A ) ⊆ · · · ⊆ F : I ∞ ( A ) ⊆ F ( A ) for any A ∈∈ C by F : I α ( A ) := { x ∈ F ( A ) | f ∗ ( x ) = 0 for all f ∈ I α ( A, B ) , B ∈∈ C } . If I and F are compatible with direct sums, then F : I ∞ ( A ) = S n ∈ N F : I n ( A ) (see Theorem 5.1), but this need not be the case in general.The functoriality of F restricts to maps I β ( A, B ) ⊗ F : I α + β ( B ) → F : I α ( A ) , f ⊗ x f ∗ ( x ) , for all α, β . In particular, F : I α is a covariant functor on C . The ideal I β actstrivially on the subquotients F : I α + β ( A ) (cid:14) F : I α ( A ) , which therefore descend tofunctors on C / I β .We may also view F as a left module over the category C and C / I α as a C -bimodule. Then F : I α corresponds to the left C -module Hom C ( C / I α , F ) .The filtration F : I α ( A ) is closely related to projective resolutions of A . Incontrast, the filtration I α F ( A ) := { f ∗ ( ξ ) ∈ F ( A ) | f ∈ I α ( B, A ) , ξ ∈ F ( B ) for some B ∈∈ C } is related to injective (co)resolutions. There is also an increasing filtration G : I n for a contravariant functor G . The filtrations I α F and G : I α will not be used inthis article.3. From projective resolutions to complementary pairs
First we refine a projective resolution by adjoining certain phantom maps. Thisyields the phantom tower over an object (see also [3]). We show that the projec-tive resolution determines this tower uniquely up to non-canonical isomorphism.There is another tower over an object, the cellular approximation tower . These twotowers are related by various commuting diagrams and exact triangles; we call thecollection of all these exact or commuting triangles the phantom castle .The goal of this section is to show that the categories h P I i and N I are comple-mentary if I is compatible with direct sums. Before we come to that, we recall thenotion of complementary pair of subcategories and define what it means for an idealto be compatible with countable direct sums. The main ingredients in the proof arethe homotopy colimits of the phantom tower and the cellular approximation tower. OMOLOGICAL ALGEBRA IN BIVARIANT K-THEORY 7
Finally, we describe a method for checking that a given localising subcategory isreflective, that is, part of a complementary pair.All results involving infinite direct sums require that the category T has countabledirect sums. Triangulated categories involving bivariant K-theory have no morethan countable direct sums because of built-in separability assumptions that makethe analysis behind the scenes work. This is why we only use countable direct sums.Of course, everything remains true if we drop the word “countable” or replace itby another cardinality constraint.A triangulated subcategory of T is called localising (more precisely, ℵ -localising )if it is closed under countable direct sums. Localising subcategories are automati-cally thick, that is, closed under direct summands (see [21]).Let I be a homological ideal in a triangulated category T . Recall that an I -projective resolution of an object A of T is a chain complex · · · δ n +1 −−−→ P n δ n −→ P n − δ n − −−−→ · · · δ −→ P δ −→ P of I -projective objects P n , augmented by a map π : P → A , such that the aug-mented chain complex is I -exact. If I = ker F for a stable homological functor F to some Abelian category A , then I -exactness means that the chain complex · · · F ( δ n +1 ) −−−−−→ F ( P n ) F ( δ n ) −−−−→ F ( P n − ) F ( δ n − ) −−−−−→ · · · F ( δ ) −−−→ F ( P ) F ( π ) −−−−→ F ( A ) is exact in A . We say that I has enough projective objects if each A ∈∈ T has suchan I -projective resolution.3.1. The phantom tower.Definition 3.1. A phantom tower over an object A of T is a diagram(3.2) A N ι / / N ι / / ◦ (cid:8)(cid:8)(cid:8) ǫ (cid:3) (cid:3) (cid:8)(cid:8)(cid:8) N ι / / ◦ (cid:8)(cid:8)(cid:8) ǫ (cid:3) (cid:3) (cid:8)(cid:8)(cid:8) N / / ◦ (cid:8)(cid:8)(cid:8) ǫ (cid:3) (cid:3) (cid:8)(cid:8)(cid:8) · · · ◦ (cid:8)(cid:8)(cid:8) (cid:4) (cid:4) (cid:8)(cid:8)(cid:8) P π [ [ P π [ [ ◦ δ o o P π [ [ ◦ δ o o P π [ [ ◦ δ o o · · · [ [ ◦ o o with I -phantom maps ι n +1 n and I -projective objects P n for n ∈ N , such that thetriangles P n π n −−→ N n ι n +1 n −−−→ N n +1 ǫ n −→ P n [1] in (3.2) are exact for all n ∈ N and the other triangles in (3.2) commute, that is, δ n +1 = ǫ n ◦ π n +1 for all n ∈ N . Notice our convention that circled arrows are mapsof degree .Since the maps δ n in the phantom tower have degree , we slightly modify ournotion of projective resolution, letting the boundary maps have degree . Lemma 3.3.
The maps δ n for n ∈ N ≥ and π in a phantom tower over A forman I -projective resolution of A . Conversely, any I -projective resolution can beembedded in a phantom tower, which is unique up to non-canonical isomorphism.A morphism f : A → A ′ lifts ( non-canonically ) to a morphism between two givenphantom towers over A and A ′ . A chain map between projective resolutions of A and A ′ extends to the phantom towers that contain these resolutions.Proof. Let P n , π , and δ n be part of a phantom tower over A . The objects P n are I -projective by definition, and δ n ◦ δ n +1 = 0 for all n ∈ N and π ◦ δ = 0 becausethese products involve two consecutive arrows in an exact triangle. Hence the maps δ n and π form a chain complex. We claim that it is I -exact. RALF MEYER
Let F be a stable homological functor with ker F = I . Recall that a chaincomplex is I -exact if and only if its F -image is exact in the usual sense by [18,Lemma 3.9]. The exact triangles in the phantom tower yield short exact sequences F ∗ +1 ( N n +1 ) F ∗ ( P n ) ։ F ∗ ( N n ) for all n ∈ N because ι n +1 n ∈ I ; here F ∗ ( A ) := F ( A [ − n ]) . Splicing these extensionsas in the definition of the Yoneda product, we get an exact chain complex. Sincethis chain complex is · · · → F ∗ +2 ( P ) → F ∗ +1 ( P ) → F ∗ ( P ) → F ∗ ( A ) → , we have got an I -exact chain complex and hence an I -projective resolution.Now let π : P → A and δ n : P n → P n − [1] for n ∈ N ≥ form an I -projectiveresolution. We recursively construct the triangles that comprise the phantom tower.To begin with, we embed π in an exact triangle P π −→ A ι −→ N ǫ −→ P [1] . Since π is I -epic, ι is an I -phantom map and ǫ is I -monic. Thus our exacttriangle yields a short exact sequence T ∗ +1 ( P, N ) T ∗ ( P, P ) ։ T ∗ ( P, A ) for any I -projective object P . In particular, this applies to P = P and showsthat δ factors uniquely as δ = ǫ ◦ π with π ∈ T ( P , N ) .We claim that π is I -epic. Let F be a defining functor for I as above. Then F ( P ) → F ( P ) → F ( A ) is exact at F ( P ) , and F ( N ) F ( P ) ։ F ( A ) is a shortexact sequence. Hence the range of F ( δ ) is isomorphic to F ( N ) . This impliesthat F ( π ) is an epimorphism, that is, π is I -epic.Thus the maps π and δ n for n ∈ N ≥ form an I -projective resolution of N . Wemay now repeat the above process and recursively construct the phantom tower.Thus any I -projective resolution embeds in a phantom tower. Furthermore, sincethe exact triangle containing a given morphism is unique up to isomorphism andthe liftings π above are unique, there is, up to isomorphism, only one phantomtower that contains a given I -projective resolution. Of course, different resolutionsyield different phantom towers.Finally, it remains to lift a morphism f : A → A ′ to a transformation between twogiven phantom towers. First we can lift f to a chain map between the I -projectiveresolutions contained in these towers (see [18]); let P n ( f ) : P n → P ′ n for n ∈ N be this chain map. It remains to construct maps N n ( f ) : N n → N ′ n that togetherwith the maps P n ( f ) intertwine the various maps in the phantom towers. Wealready have the map N ( f ) = f . The triangulated category axioms provide a map N ( f ) : N → N ′ making the diagram P P ( f ) (cid:15) (cid:15) π / / A f (cid:15) (cid:15) ι / / N N ( f ) (cid:15) (cid:15) ǫ / / P [1] P ( f )[1] (cid:15) (cid:15) P ′ π ′ / / A ′ ι ′ / / N ′ ǫ ′ / / P ′ [1] commute. We claim that N ( f ) ◦ π = π ′ ◦ P ( f ) . As above, we get short exactsequences T ∗ +1 ( P , N ′ ) T ∗ ( P , P ′ ) ։ T ∗ ( P , A ′ ) . Hence it suffices to check ǫ ′ ◦ N ( f ) ◦ π = ǫ ′ ◦ π ′ ◦ P ( f ) = δ ′ ◦ P ( f ) . But this istrue because ǫ ′ ◦ N ( f ) = P ( f ) ◦ ǫ and the maps P n ( f ) form a chain map. Thusthe map N ( f ) has all required properties. Iterating this construction, we get the OMOLOGICAL ALGEBRA IN BIVARIANT K-THEORY 9 maps N n ( f ) for all n ∈ N . By the way, they need not be unique even if the maps P n ( f ) are fixed. (cid:3) The following definition formalises an important property of the maps ι n +1 n in aphantom tower. Definition 3.4.
Let I ⊆ T be an ideal. Let A, B ∈∈ T . We call f ∈ I ( A, B ) I -versal if, for any C ∈∈ T , any g ∈ I ( A, C ) factors as g = h ◦ f for some h ∈ T ( B, C ) : A f / / g (cid:31) (cid:31) @@@@@@@ B h ∃ (cid:15) (cid:15) C We do not require this factorisation to be unique.Since I is an ideal, any map of the form h ◦ f belongs to I . Lemma 3.5.
The maps ι n +1 n in a phantom tower are I -versal for all n ∈ N .Proof. Let f ∈ I ( N n , B ) . Since P n is I -projective, I ∗ ( P n , B ) = 0 . Thus f ◦ π n = 0 .This forces f to factor through ι n +1 n because T ∗ ( , B ) is cohomological. (cid:3) Lemma 3.6.
Let I and I be ideals in a triangulated category. If f ∈ I ( B, C ) and f ∈ I ( A, B ) are I - and I -versal maps, respectively, then f ◦ f : A → C is I ◦ I -versal.Proof. Let h ∈ I ◦ I ( A, D ) , write h = h ◦ h with h ∈ I and h ∈ I . Usingversality of f and f , we find the maps h ′ and h ′ in the following diagram: A f / / h (cid:31) (cid:31) @@@@@@@@ B f / / h ′ ∃ (cid:15) (cid:15) C h ′ ∃ (cid:15) (cid:15) • h / / D. Thus h factors through f ◦ f as required. (cid:3) As a consequence, the maps ι n + kn := ι n + kn + k − ◦ · · · ◦ ι n +1 n : N n → N n + k in a phantom tower are I k -versal for all n, k ∈ N . Lemma 3.7.
A map f : A → B is I k -versal if and only if I k ( A, C ) = range (cid:0) f ∗ : T ( B, C ) → T ( A, C ) (cid:1) for all C ∈∈ T .Let f : A → B be I k -versal. If F : T → Ab is homological, then F : I k ( A ) = ker (cid:0) f ∗ : F ( A ) → F ( B ) (cid:1) ; if G : T op → Ab is cohomological, then I k G ( A ) = range (cid:0) f ∗ : G ( B ) → G ( A ) (cid:1) . Proof.
This follows immediately from the definitions. (cid:3)
As a consequence, we can compute the filtrations F : I k ( A ) and I k G ( A ) ofSection 2.2 from the phantom tower. The phantom castle.
Now we extend the phantom tower to the phantomcastle, which contains among other things the cellular approximation tower. Westart with a phantom tower over some object A ∈∈ T . Let ι n := ι nn − ◦ · · · ◦ ι : A = N → N n and embed ι n in an exact triangle(3.8) ˜ A n α n −−→ A ι n −→ N n γ n −→ ˜ A n [1] . The octahedral axiom relates the mapping cones ˜ A n +1 , P n , and ˜ A n of the maps ι n +1 , ι n +1 n , and ι n because ι n +1 = ι n +1 n ◦ ι n (see [21, Proposition I.4.6] or [16, PropositionA.1]). More precisely, the octahedral axiom allows us to choose maps(3.9) ˜ A n α n +1 n −−−→ ˜ A n +1 σ n −−→ P n κ n −−→ ˜ A n [1] , such that this triangle is exact and the following diagram commutes:(3.10) N n ι n +1 n / / ◦ γ n (cid:15) (cid:15) N n +1 ◦ GGGG ǫ n GGGG ◦ γ n +1 (cid:15) (cid:15) ˜ A n α n +1 n / / α n " " EEEEEEEEE ˜ A n +1 α n +1 (cid:15) (cid:15) σ n / / P nπ n (cid:15) (cid:15) ◦ κ n / / ˜ A n A ι n / / ι n +1 HHHHHHHHH N n ◦ yyyy γ n < < yyyy ι n +1 n (cid:15) (cid:15) N n +1 In addition, we can achieve that the triangle(3.11) N n [ − → ˜ A n +1 (cid:0) α n +1 σ n (cid:1) −−−−−→ A ⊕ P n ( ι n ,π n ) −−−−−→ N n is exact, that is, the square in the middle of (3.10) is homotopy Cartesian and thediagonal of the top right square provides its differential. Lemma 3.12.
The object ˜ A n is I n -projective for each n ∈ N .Proof by induction on n . The case n = 0 is clear. Since P n ∈∈ P I for all n ∈ N ,the exact triangles (3.9) and Proposition 2.2 provide the induction step. (cid:3) Furthermore, the map α n : ˜ A n → A is I n -epic because ι n ∈ I n , so that it is thefirst step of an I n -projective resolution of A . This provides another explanationwhy the map ι n is I -versal (compare Lemma 3.5). Remark . The cone of the map ι m + km is I k -projective for all m, k ∈ N by asimilar argument. Hence A = N ι k −→ N k ι kk −−→ N k ι k k −−→ N k → · · · together with the exact triangles that contain the maps ι jk + kjk is an I k -phantomtower and hence yields an I k -projective resolution by Lemma 3.3.As a result, an I -phantom tower determines I k -phantom towers for all k ∈ N . Definition 3.14.
The sequence of exact triangles (3.9) is called the cellular ap-proximation tower over A . OMOLOGICAL ALGEBRA IN BIVARIANT K-THEORY 11
The motivation for our terminology is the following. If A has a projective res-olution of finite length, then we can choose a phantom tower with P n = 0 for n ≫ . Suppose, in addition, that A belongs to the thick triangulated subcategorygenerated by P I . Then the proof of Proposition 4.10 yields N n = 0 for n ≫ .The exact triangles (3.8) mean that the maps α n : ˜ A n → A become invertible for n ≫ , that is, ˜ A n ∼ = A . Therefore, we think of the objects N n as “obstructions”that should get smaller for n → ∞ , and of the objects ˜ A n as better and betterapproximations to A . They are called “ P I -cellular” because they are constructedout of I -projective objects – the cells – by iterated exact triangles. Definition 3.15. A phantom castle over A is a sequence of objects N n , P n , ˜ A n with maps ι n +1 n , π n , ǫ n , ι n , α n , γ n , σ n , κ n such that the triangles (4.1), (3.8), (3.9),and (3.11) are exact and the diagram (3.10) commutes.We will use most of the information encoded in this definition to identify thespectral sequences generated by the phantom tower and the cellular approximationtower; only the commutativity of the square in the middle of (3.10) and the exactsequence (3.11) seem irrelevant in the following.3.3. Complementary pairs of subcategories and localisation.
We call twothick subcategories L and N of T complementary if T ∗ ( L, N ) = 0 for all L ∈∈ L , N ∈∈ N and, for any A ∈∈ T , there is an exact triangle L → A → N → L [1] with L ∈∈ L and N ∈∈ N (see [16, Definition 2.8]). Similar situations have beenstudied by various authors, under various names, such as localisation pairs, stablet-structures, torsion pairs; a complementary pair is equivalent to a localisationfunctor L on T , where L is the class of L -local objects and N is the class of L -acyclicobjects.The following assertions are contained in [16, Proposition 2.9]. Let ( L , N ) becomplementary. Then the exact triangle L → A → N → L [1] with L ∈∈ L and N ∈∈ N is unique and functorial, and the resulting functors L : T → L and N : T → N mapping A to L and N , respectively, are left adjoint to the embeddingfunctor L → T and right adjoint to the embedding functor N → T , respectively.That is, the subcategory N is reflective and L is coreflective. The composite functors L → T → T / N and N → T → T / L are equivalences of categories.Conversely, let N ⊆ T be a reflective subcategory and let N : T → N be the leftadjoint of the embedding functor N → T . Let L = { A ∈∈ T | N ( A ) = 0 } be the left orthogonal complement of N . Then ( L , N ) is a complementary pairof subcategories, and L is the only possible partner for N . Thus complementarypairs are essentially the same as reflective subcategories. Dually, a subcategory L is coreflective if and only if it is part of a complementary pair ( L , N ) , and the onlycandidate for N is the right orthogonal complement of L .If F : T → C is a covariant functor, then its localisation L F with respect to N is defined by L F := F ◦ L , where L : T → L is the right adjoint of the embedding L → T . The natural maps L ( A ) → A provide a natural transformation L F ⇒ F .If G : T op → C is a contravariant functor, then the localisation G ◦ L is denotedby R G . It comes together with a natural transformation G ⇒ R G .This localisation process is an important tool to construct functors. Special casesare derived functors in homological algebra and the domain of the Baum–Connesassembly map (see [16]).Although the definition of a complementary pair is symmetric, the subcategories L and N have a rather different nature in most examples. Usually, one of them– here it is always N – is defined directly and the other one is only described by generators. This makes it hard to tell which objects it contains and to find theexact triangles needed for complementarity.Here homological ideals help. Let I be a homological ideal with enough projec-tive objects in a triangulated category T . Let h P I i be the localising subcategorygenerated by P I , that is, the smallest localising subcategory of T that contains P I .Since the name “projective” is already taken, we call objects of h P I i P I -cellular .We have P n I ⊆ h P I i for all n ∈ N ∪ {∞ , ∞ , . . . } by Propositions 2.2 and 2.3. Definition 3.16.
Let N I be the full subcategory of I -contractible objects, that is,objects N with id N ∈ I ( N, N ) .An object N is I -contractible if and only if → N is an I -projective resolution.Thus all I -derived functors vanish on N I .Now the following question arises: is the pair of subcategories ( h P I i , N I ) com-plementary? It is evident that T ( P, N ) = 0 if P ∈ P I and N ∈ N I . This extendsto P ∈ h P I i because the left orthogonal complement of N I is localising. This isthe easy half of the definition of a complementary pair. The other, non-trivial halfrequires an additional condition on the ideal I .3.4. Compatibility with direct sums.Definition 3.17.
An ideal I is called compatible with countable direct sums if, forany countable family ( A i ) i ∈ I of objects of T , the canonical isomorphism T (cid:18)M i ∈ I A i , B (cid:19) ∼ = Y i ∈ I T ( A i , B ) restricts to an isomorphism I (cid:0)L i ∈ I A i , B (cid:1) ∼ = Q i ∈ I I ( A i , B ) .An ideal I is compatible with countable direct sums if and only if the followingholds: given countable families of objects ( A i ) , ( B i ) and maps f i ∈ I ( A i , B i ) for i ∈ I , we have L f i ∈ I (cid:0)L A i , L B i (cid:1) .Recall that direct sums of exact triangles are again exact (see [21]) and thatthe ideal determines and is determined by the classes of I -epimorphisms or of I -monomorphisms. Therefore, I is compatible with direct sums if and only if directsums of I -monomorphisms are again I -monomorphisms, if and only if direct sumsof I -epimorphisms are again I -epimorphisms.Moreover, if I is compatible with countable direct sums, then a direct sum of I -equivalences is again an I -equivalence, and N I is a localising subcategory of T .Moreover, a direct sum of phantom castles over A i is a phantom castle over L A i . Example . Let F be a stable homological functor or an exact functor to anothertriangulated category, and suppose that F commutes with countable direct sums.Then ker F is a homological ideal and compatible with countable direct sums.This example is, in fact, already the most general case: Proposition 3.19.
Let T be a triangulated category with countable direct sums andlet I be a homological ideal in T . Let F : T → A I T be a universal I -exact stablehomological functor. The ideal I is compatible with countable direct sums if andonly if the Abelian category A I T has exact countable direct sums and the functor F : T → A I T commutes with countable direct sums.Proof. One direction is just the assertion in Example 3.18. The other directionrequires some description of the universal functor F . We use the description in [18],which starts with the homotopy category Ho( T ) of chain complexes with entries in T .Since T has countable direct sums, so has Ho( T ) . The I -exact chain complexes forma thick subcategory E of Ho( T ) ; it is closed under countable direct sums because I OMOLOGICAL ALGEBRA IN BIVARIANT K-THEORY 13 is compatible with countable direct sums. Hence the localisation
Ho( T ) / E still hascountable direct sums.The Abelian approximation A I T is equivalent to the heart of a canonical trun-cation structure on Ho( T ) / E described in [18] and consists of chain complexes thatare exact in degrees not equal to . The universal functor F is the obvious one,viewing an object of T as a chain complex supported in degree . It is evident thatthe subcategory A I T ⊆ Ho( T ) / E is closed under countable direct sums. Countabledirect sums of extensions in A I T remain extensions because the analogous assertionholds for direct sums of exact triangles in any triangulated category (see [21]) andextensions in the heart are related to exact triangles in the ambient triangulatedcategory. Clearly, the functor T → A I T preserves countable direct sums. (cid:3) Example . The ideal of finite rank operators on the category of vector spacesis an ideal that is not compatible with countable direct sums.3.5.
Complementarity and structure of cellular objects.
The results in thissection generalise results of Apostolos Beligiannis (see [3, Theorem 6.5], [3, Corol-lary 5.12]) in the case where P I is generated by a single compact object. Theorem 3.21.
Let T be a triangulated category with countable direct sums, andlet I be a homological ideal in T with enough projective objects. Suppose that I is compatible with countable direct sums. Then the pair of localising subcategories ( h P I i , N I ) in T is complementary. We will present two independent proofs, one using phantom towers, the othercellular approximation towers. Both require homotopy colimits:
Definition 3.22.
Let ( D n , ϕ n +1 n ) be an inductive system in T . Define the shift S : M D n → M D n , S | D n : D n ϕ n +1 n −−−→ D n +1 ⊆ M D n . The homotopy colimit ho- lim −→ ( D n , ϕ n +1 n ) is the third leg in the exact triangle L D n id − S / / L D n / / ho- lim −→ ( D n , ϕ n +1 n ) / / L D n [1] . Recall that id − S determines this triangle uniquely up to isomorphism. Proof of Theorem 3.21.
Since the class of A ∈∈ T with T ∗ ( A, B ) = 0 for all B ∈∈ N I is localising, we have T ∗ ( A, B ) = 0 if A ∈∈ h P I i and B ∈∈ N I . It remains toconstruct, for each A ∈∈ T , an exact triangle ˜ A → A → N → ˜ A [1] with N ∈∈ N I and ˜ A ∈∈ h P I i .Construct a phantom castle over A and let N := ho- lim −→ ( N n , ι n +1 n ) be the ho-motopy colimit of the phantom tower. We also use the homotopy colimit of theconstant inductive system ( A, id A ) . This is just A because of the split exact triangle(3.23) M A id − S −−−→ M A ∇ −→ A −→ M A [1] , where ∇ is the codiagonal map. By [2, Proposition 1.1.11] (and a rotation), we canfind ˜ A and the dotted arrows in the following diagram(3.24) L ˜ A n / / L α n (cid:15) (cid:15) L ˜ A n / / L α n (cid:15) (cid:15) ˜ A ◦ / / (cid:15) (cid:15) L ˜ A n L α n (cid:15) (cid:15) L A id − S / / L ι n (cid:15) (cid:15) L A L ι n (cid:15) (cid:15) ∇ / / A ◦ / / (cid:15) (cid:15) L A L ι n (cid:15) (cid:15) L N n id − S / / ◦ L γ n (cid:15) (cid:15) L N n ◦ L γ n (cid:15) (cid:15) / / N ◦ / / ◦ (cid:15) (cid:15) − L N n ◦− L γ n (cid:15) (cid:15) L ˜ A n / / L ˜ A n / / ˜ A [1] ◦ / / L ˜ A n so that the rows and columns are exact triangles and the squares commute exceptfor the one marked with a minus sign, which anti-commutes.Lemma 3.12 yields ˜ A n ∈∈ P n I for all n ∈ N . Hence L ˜ A n ∈∈ P ∞ I ⊆ h P I i byProposition 2.3. The exactness of the first row in (3.24) implies ˜ A ∈∈ h P I i . Weclaim that N ∈∈ N I . Hence the third column in (3.24) is the kind of exact trianglewe need for h P I i and N I to be complementary.Let F be a stable homological functor with ker F = I . We must show F ( N ) = 0 .The map S factors through L ι n +1 n ; this map belongs to I = ker F because I iscompatible with direct sums. Hence F ( id − S ) = F ( id ) is invertible. By a longexact sequence, this implies F ( N ) = 0 , that is, N ∈∈ N I . (cid:3) Suppose from now on that we are in the situation of Theorem 3.21. Since h P I i and N I are complementary, there is a unique exact triangle ˜ A → A → N → ˜ A [1] with I -contractible N and P I -cellular ˜ A ; we call ˜ A the P I -cellular approximation of A . Even more, ˜ A and N depend functorially on A , so that we get two functors L : T → h P I i and N : T → N I .The proof of Theorem 3.21 above provides an explicit model for N ( A ) : it is thehomotopy colimit of the phantom tower of A . Proposition 3.25.
Let A ∈∈ T and construct a phantom castle over A . Then L ( A ) = ˜ A is the homotopy colimit of the cellular approximation tower ( ˜ A n ) n ∈ N . This does not yet follow from (3.24) because we cannot control the dotted maps.
Proof.
Let ˜ A := ho- lim −→ ( ˜ A n , α n +1 n ) . We compare the exact triangle that definesthe homotopy colimit ˜ A with the triangle (3.23). The triangulated category axiomsprovide f ∈ T ( ˜ A, A ) that makes the following diagram commute:(3.26) L ˜ A n id − S / / L α n (cid:15) (cid:15) L ˜ A n / / L α n (cid:15) (cid:15) ˜ A f (cid:15) (cid:15) ◦ / / L ˜ A n L α n (cid:15) (cid:15) L A id − S / / L A ∇ / / A ◦ / / L A. We claim that f is an I -equivalence. Equivalently, the cone of f is I -contractible,so that the mapping cone triangle for f has entries in h P I i and N I ; this impliesthat L ( A ) = ˜ A .Let F be a stable homological functor with ker F = I . We check that F ( f ) isinvertible. The direct sum of the triangles (3.8) for n ∈ N is again an exact triangle. OMOLOGICAL ALGEBRA IN BIVARIANT K-THEORY 15
On the long exact homology sequence · · · −→ F m +1 (cid:16)M N n (cid:17) −→ F m (cid:16)M ˜ A n (cid:17) −→ F m (cid:16)M A (cid:17) −→ F m (cid:16)M N n (cid:17) −→ · · · for this exact triangle, consider the operator induced by id − S on each entry. Since I is compatible with direct sums, the shift map S on L N n is a phantom map, sothat F ( id − S ) acts identically on F (cid:0)L N n (cid:1) . On F (cid:0)L A (cid:1) , the map id − S induces asplit monomorphism with cokernel F ( ∇ ) : F (cid:0)L A (cid:1) → F ( A ) . Now a diagram chaseshows that the map on F (cid:0)L ˜ A n (cid:1) induced by id − S is injective and has the cokernel F (cid:16)M ˜ A n (cid:17) F (cid:0) L α n (cid:1) −−−−−−→ F (cid:16)M A (cid:17) F ( ∇ ) −−−→ F ( A ) . Comparing the long exact homology sequences for the two rows in (3.26), we con-clude that F ( f ) is indeed invertible. (cid:3) We have proved Proposition 3.25 by constructing an I -equivalence between anarbitrary object of T and the homotopy colimit of its cellular approximation tower.This provides another, independent proof of Theorem 3.21. Since the exact triangle ˜ A → A → N → ˜ A [1] with ˜ A ∈∈ h P I i and N ∈∈ N I is unique up to isomorphism,both proofs construct the same exact triangle. The first proof shows that N isthe homotopy colimit of the phantom tower, the second one shows that ˜ A is thehomotopy colimit of the cellular approximation tower. We conclude, therefore, thatthe object ˜ A in (3.24) is the homotopy colimit of the phantom tower and that themap ˜ A → A in (3.24) agrees with the map f from (3.26). Theorem 3.27.
Let T be a triangulated category with countable direct sums, andlet I be a homological ideal in T that is compatible with countable direct sums andhas enough projective objects. Let A ∈∈ T and construct a phantom castle over A .The following are equivalent: (1) A is P I -cellular, that is, A ∈∈ h P I i ; (2) A is isomorphic to the homotopy colimit of its cellular approximation tower; (3) A is isomorphic to the homotopy colimit of an inductive system ( P n , ϕ n ) with P n ∈∈ S k ∈ N P k I for all n ∈ N ; (4) A is I ∞ -projective.As a consequence, I ∞ = I n ∞ for all n ≥ .Proof. Since L ( A ) ∼ = A if and only if A ∈∈ h P I i , Proposition 3.25 yields the equiva-lence of (1) and (2). The implication (2) = ⇒ (3) is trivial: the cellular approximationtower provides an inductive system of the required kind.We check that (3) implies (4). Let ( P n , ϕ n ) be an inductive system as in (3).First, Proposition 2.3 shows that L P n is I ∞ -projective. Then Proposition 2.2shows that the homotopy colimit is I ∞ -projective.Propositions 2.3 and 2.2 show recursively that all I α -projective objects belongto h P I i for n = 0 , , , , . . . , ∞ , · ∞ . Hence (4) implies (1), so that all fourconditions are equivalent.Finally, since P ∞ I = h P I i , it follows from Proposition 2.2 that the powers I n ∞ for n ≥ have the same projective objects. Therefore, they are all equal. (cid:3) Proposition 3.28.
The P I -cellular approximation functor L : T → h P I i mapsa phantom castle over A ∈∈ T to a phantom castle over L ( A ) . A morphism f ∈ T ( A, B ) belongs to I α for some α if and only if L ( f ) ∈ T (cid:0) L ( A ) , L ( B ) (cid:1) does. Proof.
Since L is an exact functor, it preserves the commuting diagrams and exacttriangles required for a phantom castle. It also maps P I to itself because L ( B ) ∼ = B for all B ∈∈ h P I i . It remains to check that L ( f ) ∈ I α (cid:0) L ( B ) , L ( B ′ ) (cid:1) if and onlyif f ∈ I α ( B, B ′ ) . Let F be a stable homological functor with I α = ker F . Then F ( N ) = 0 for all N ∈∈ N I . Therefore, F descends to the localisation T / N I ;equivalently, the natural transformation F ◦ L ⇒ F is an isomorphism. In particular, F ( f ) = 0 if and only if F (cid:0) L ( f ) (cid:1) = 0 . (cid:3) As a result, it makes almost no difference whether we work in T or T / N I . Wework in T most of the time and allow N I to be non-trivial in order to formulateTheorem 3.21.The direct sums L ˜ A n in (3.26) are I ∞ -projective by Proposition 2.3. The map ∇ ◦ L α n : L ˜ A n → A is I ∞ -epic because it is I n -epic for all n ∈ N . We mayreplace A by ˜ A in this statement by Proposition 3.28. Thus the top row in (3.26)is an I ∞ -exact triangle. This means that the chain complex · · · → → M ˜ A n id − S −−−→ M ˜ A n → ˜ A is I ∞ -exact and hence an I ∞ -projective resolution of ˜ A . Once again, Propo-sition 3.28 allows us to replace ˜ A by A in this statement, that is, we get an I ∞ -projective resolution of length (3.29) · · · → → M ˜ A n id − S −−−→ M ˜ A n → A. This will allow us to analyse the convergence of the ABC spectral sequence.3.6.
Complementarity via partially defined adjoints.
Suppose that we aregiven a thick subcategory N of a triangulated category T and that we want to useTheorem 3.21 to show that it is reflective, that is, there is another thick subcate-gory L such that ( L , N ) is complementary.To have a chance of doing so, T must have countable direct sums, and thesubcategory N must be localising, that is, closed under countable direct sums:this happens whenever Theorem 3.21 applies. By [16, Proposition 2.9], the onlycandidate for L is the left orthogonal complement L := { A ∈∈ T | T ( A, N ) = 0 for all N ∈∈ N } of N , which is another localising subcategory.The starting point of our method is a stable additive functor F : T → C with N = N F := { A ∈∈ T | F ( A ) = 0 } . This functor yields a stable ideal I F := ker F . In applications, F is either a stablehomological functor to a stable Abelian category or an exact functor to anothertriangulated category; in either case, the ideal ker F is homological and N F is theclass of all I F -contractible objects. In addition, we assume F to commute withcountable direct sums , so that I F is compatible with countable direct sums.In order to apply Theorem 3.21, it remains to prove that there are enough I F -projective objects in T . Then the pair of subcategories ( h P I F i , N ) is comple-mentary. For a good choice of F , this may be much easier than proving directlythat ( L , N ) is complementary. The choice in the following example never helps.But often, there is another choice for F that does. Example . The localisation functor T → T / N is a possible choice for F – thatis, it has the right kernel on objects – but it should be a bad one because it tells usnothing new about N . In fact, the ker F -projective objects are precisely the objectsof L , so that we have not gained anything. OMOLOGICAL ALGEBRA IN BIVARIANT K-THEORY 17
We now discuss a sufficient condition for enough projective objects from [18].The left adjoint of F : T → C is defined on an object A ∈∈ C if the functor B C (cid:0) A, F ( B ) (cid:1) on T is representable, that is, there is an object F ⊢ ( A ) of T and anatural isomorphism T ( F ⊢ ( A ) , B ) ∼ = C (cid:0) A, F ( B ) (cid:1) for all B ∈∈ T . We say that F ⊢ is defined on enough objects if, for any object B of C there is an epimorphism B ′ → B such that F ⊢ is defined on B ′ .The following theorem asserts that I F has enough projective objects if F ⊢ isdefined on enough objects. The statement is somewhat more involved becauseit is often useful to shrink the domain of definition of F ⊢ to a sufficiently bigsubcategory PC . Theorem 3.31.
Let T , C , and F be as above, that is, T is a triangulated categorywith countable direct sums, C is either a stable Abelian category or a triangulatedcategory, and F : T → C is a stable functor commuting with countable direct sumsand either homological ( if C is Abelian ) or exact ( if C is triangulated ) . Let I F :=ker F and let N F be the class of F -contractible objects as above.Let PC ⊆ C be a subcategory with two properties: first, for any A ∈∈ T , thereexists an epimorphism P → F ( A ) with P ∈∈ PC ; secondly, the left adjoint func-tor F ⊢ of F is defined on PC , that is, for each P ∈ PC , there is an object F ⊢ ( P ) in T with T ( F ⊢ ( P ) , B ) ∼ = T (cid:0) P, F ( B ) (cid:1) naturally for all B ∈∈ T .Then I F has enough projective objects, the subcategory N F is reflective, and thepair of localising subcategories (cid:0) h F ⊢ ( PC ) i , N F (cid:1) is complementary.Proof. [18, Proposition 3.37] shows that I F has enough projective objects and thatany projective object is a direct summand of F ⊢ ( P ) for some P ∈ PC . NowTheorem 3.21 yields the assertions. (cid:3) Theorem 3.31 is non-trivial even if N F contains only zero objects, that is, if F ( A ) = 0 implies A = 0 . Then it asserts h F ⊢ ( PC ) i = T .In the situation of Theorem 3.31, we also understand how objects of h F ⊢ ( PC ) i are to be constructed from the building blocks in F ⊢ ( PC ) .Let C ⋆ C for subcategories C , C ⊆ T be the subcategory of all objects A for which there is an exact sequence A → A → A → A [1] with A ∈∈ C and A ∈∈ C . We abbreviate P F := F ⊢ ( PC ) and recursively define P ⋆nF for n ∈ N by P ⋆ F := { } and P ⋆nF := P ⋆n − F ⋆ P F for n ≥ . Theorem 3.32.
In the situation of Theorem , any object of h F ⊢ ( PC ) i is ahomotopy colimit of an inductive system ( A n ) n ∈ N with A n ∈ P ⋆nF .Proof. This follows from Theorem 3.27. But an extra observation is needed herebecause we do not adjoin direct summands of objects in the definition of the sub-categories P ⋆nF , so that they do not necessarily contain all I nF -projective objects.There is an I F -projective resolution with entries in P F , which we embed in aphantom castle. The resulting cellular approximation tower satisfies ˜ A n ∈∈ P ⋆nF ,so that Theorem 3.27 yields an inductive system of the required form. (cid:3) We have considered two cases above: homological and exact functors. For ho-mological functors with values in the category of Abelian groups, our results wereobtained previously by Apostolos Beligiannis [3]. Let F : T → Ab Z be a stablehomological functor that commutes with direct sums. Suppose that F is defined onsufficiently many objects. Then there must be a surjective map A ։ Z for which F ⊢ ( A ) is defined. Since Z is projective, Z is a retract of A . Since we assume T tohave direct sums, idempotent morphisms in T have range objects. Thus F ⊢ ( Z ) isdefined as well. By definition, F ⊢ ( Z ) is a representing object for F . Conversely,if F is representable, then F ⊢ can be defined on all free Abelian groups. Hence the adjoint F ⊢ is defined on sufficiently many objects if and only if F is representable.Furthermore, we can take P F to be the set of all direct sums of translated copies ofthe representing object F ⊢ ( Z ) . The assumption that F commute with direct sumsmeans that F ⊢ ( Z ) is a compact object.Summing up, if F is a stable homological functor to Ab Z , then our methods applyif and only if F ( A ) ∼ = T ∗ ( D, A ) for a compact generator D of T . This situation isconsidered already in [3].4. The ABC spectral sequence
When we apply a homological or cohomological functor to the phantom tower, weget first an exact couple and then a spectral sequence. We call it the ABC spectralsequence after Adams, Brinkmann, and Christensen. Its second page only involvesderived functors. The higher pages can be described in terms of the phantom tower,but are more complicated. It is remarkable that the ABC spectral sequence is well-defined and functorial on the level of triangulated categories, that is, all the higherboundary maps are uniquely determined and functorial without introducing finerstructure like, say, model categories.Several results in this section are already know to the experts or can be ex-tracted from [3, 4, 7]. We have included them, nevertheless, to give a reasonablyself-contained account.4.1.
A spectral sequence from the phantom tower.
We are going to constructexact couples out of the phantom tower, extending results of Daniel Christensen [7].We fix A ∈∈ T and a phantom tower (3.2) over A . In addition, we let P n := 0 , N n := A, and ι n +1 n := id A for n < . Thus the triangles(4.1) P n π n −−→ N n ι n +1 n −−−→ N n +1 ǫ n −→ P n [1] are exact for all n ∈ Z . Of course, ι n +1 n rarely belongs to I for n < .Let F : T → Ab be a homological functor. Define bigraded Abelian groups D := X p,q ∈ Z D pq , D pq := F p + q +1 ( N p +1 ) ,E := X p,q ∈ Z E pq , E pq := F p + q ( P p ) , and homogeneous group homomorphisms D i / / D j (cid:4) (cid:4) (cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:10) E k Z Z i pq := ( ι p +2 p +1 ) ∗ : D p,q → D p +1 ,q − , deg i = (1 , − ,j pq := ( ǫ p ) ∗ : D p,q → E p,q , deg j = (0 , ,k pq := ( π p ) ∗ : E p,q → D p − ,q , deg k = ( − , . Since F is homological and the triangles (4.1) are exact, the chain complexes · · · / / F m ( P n ) π n ∗ / / F m ( N n ) ι n +1 n ∗ / / F m ( N n +1 ) ǫ n ∗ / / F m − ( P n ) / / · · · are exact for all m ∈ Z . Hence the data ( D, E, i, j, k ) above is an exact couple (see[15, Section XI.5]).We briefly recall how an exact couple yields a spectral sequence, see also [15, page336–337] or [4]. The first step is to form derived exact couples . Let D r := i r − ( D ) ⊆ D, E r := k − ( D r ) (cid:14) j (ker i r − ) , OMOLOGICAL ALGEBRA IN BIVARIANT K-THEORY 19 for all r ≥ . Let i ( r ) : D r → D r be the restriction of i ; let k ( r ) : E r → D r beinduced by k : E → D ; and let j ( r ) : D r → E r be induced by the relation j ◦ i − r .It is shown in [15] that E r +1 ∼ = H ( E r , d ( r ) ) for all r ∈ N , where d ( r ) = j ( r ) k ( r ) ; themap d ( r ) has bidegree ( − r, r − . We call this spectral sequence the ABC spectralsequence for F and A .We are going to describe the derived exact couples explicitly. First, we claimthat(4.2) D r +1 p − ,q ∼ = F p + q ( A ) for p ≤ , F p + q ( A ) (cid:14) F p + q : I p ( A ) for ≤ p ≤ r , F p + q ( N p − r ) (cid:14) F p + q : I r ( N p − r ) for r ≤ p .By definition, D r +1 p − ,q for r ∈ N is the range of the map i r : D p − − r,q + r → D p − ,q .This is the identity map on F p + q ( A ) for p ≤ , the map ι p ∗ : F p + q ( A ) → F p + q ( N p ) for ≤ p ≤ r , and the map ( ι pp − r ) ∗ : F p + q ( N p − r ) → F p + q ( N p ) for r ≤ p . Now recallthat the maps ι nm are I n − m -versal for all n ≥ m ≥ and use Lemma 3.7. Proposition 4.3.
Let ≤ r < ∞ . Then we have E r +1 pq ∼ = 0 for p ≤ − ; for ≤ p ≤ r , there is an exact sequence → F p + q +1 ( N p ) F p + q +1 : I r +1 ( N p ) ( ι p +1 p ) ∗ −−−−−→ F p + q +1 ( N p +1 ) F p + q +1 : I r ( N p +1 ) → E r +1 pq → F p + q : I p +1 ( A ) F p + q : I p ( A ) → for p ≥ r , there is an exact sequence → F p + q +1 ( N p ) F p + q +1 : I r +1 ( N p ) ( ι p +1 p ) ∗ −−−−−→ F p + q +1 ( N p +1 ) F p + q +1 : I r ( N p +1 ) → E r +1 pq → F p + q : I r +1 ( N p − r ) F p + q : I r ( N p − r ) → . Finally, for r = ∞ , let Bad pq := F q ( N p ) . [ r ∈ N F q : I r ( N p ); then we get E ∞ pq ∼ = 0 for p ≤ − , and exact sequences → Bad p,p + q +1 → Bad p +1 ,p + q +1 → E ∞ pq → F p + q : I p +1 ( A ) F p + q : I p ( A ) → . Therefore, if S r ∈ N F q : I r ( N p ) = F q ( N p ) for all p ∈ N , then E ∞ pq ∼ = F p + q : I p +1 ( A ) F p + q : I p ( A ) , that is, the ABC spectral sequence converges towards F m ( A ) , and the induced in-creasing filtration on the limit is (cid:0) F m : I r ( A ) (cid:1) r ∈ N .Proof. We have E r +1 pq ∼ = 0 for p ≤ − because already E pq = E pq = 0 . Let p ≥ .We use (4.2) and the exactness of the derived exact couple ( D r +1 , E r +1 ) to com-pute E r +1 by an extension involving the kernel and cokernel of the restriction of i to D r +1 . Since ι m +1 m is I -versal, x ∈ F ( N m ) satisfies ( ι m +1 m ) ∗ ( x ) ∈ F : I r ( N m +1 ) ifand only if x ∈ F : I r +1 ( N m ) . Plugging this into the extension that describes E r +1 ,we get the assertion, at least for finite r .The case r = ∞ is similar. Now(4.4) E ∞ := \ r ∈ N k − ( i r D ) . [ r ∈ N j (ker i r ) . The injectivity of the map
Bad p,p + q +1 → Bad p +1 ,p + q +1 follows from the exactnessof colimits of Abelian groups. Using i ( D ) = ker j , ker i = k ( E ) , and (4.4), we get ashort exact sequence(4.5) → D pq . (cid:16) i ( D p − ,q +1 ) + [ ker i r (cid:17) → E ∞ pq → D p − ,q ∩ ker i ∩ \ i r ( D ) → the first map is induced by j , the second one by k .The intersection T i r ( D ) is described by (4.2) for r = ∞ , so that the third casein (4.2) is missing. Hence the quotient in (4.5) is ker i ∩ \ i r ( D ) ∼ = F p + q : I p +1 ( A ) (cid:14) F p + q : I p ( A ) . The versality of the maps ι nm yields S ker i r ∩ D p − ,q = S r F p + q : I r ( N p ) . Hencethe kernel in (4.5) is Bad p +1 ,p + q +1 / Bad p,p + q +1 . If Bad = 0 , then the groups E ∞ pq for p + q = m are the subquotients of the filtration F m : I p ( A ) on F m ( A ) . Moreover,since Bad m = F m ( A ) . S p ∈ N F m : I p ( A ) , our hypothesis includes the statementthat F m ( A ) = S p ∈ N F m : I p ( A ) . (cid:3) Dual constructions apply to a cohomological functor G : T op → Ab . Equa-tion (4.1) yields a sequence of exact chain complexes · · · / / G m − ( P n ) ǫ ∗ n / / G m ( N n +1 ) ( ι n +1 n ) ∗ / / G m ( N n ) π ∗ n / / G m ( P n ) / / · · · . Therefore, the following defines an exact couple: ˜ D pq := G p + q +1 ( N p +1 ) , ˜ E pq := G p + q ( P p ) , ˜ D i / / ˜ D j (cid:5) (cid:5) (cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11) ˜ E k Y Y i pq := ( ι p +1 p ) ∗ : ˜ D p,q → ˜ D p − ,q +1 , deg i = ( − , ,j pq := π ∗ p +1 : ˜ D p,q → ˜ E p +1 ,q , deg j = (1 , ,k pq := ǫ ∗ p : ˜ E p,q → ˜ D p,q , deg k = (0 , . Again we form derived exact couples ( ˜ D r , ˜ E r , i r , j r , k r ) , and ( ˜ E r , d r ) with d r := j r k r is a spectral sequence. The map d r has bidegree ( r, − r ) . We call this spectralsequence the ABC spectral sequence for G and A .We can describe the derived exact couples as above. To begin with,(4.6) ˜ D p − ,qr +1 = G p + q ( A ) for p ≤ , I p G p + q ( A ) for ≤ p ≤ r , I r G p + q ( N p − r ) for r ≤ p . Proposition 4.7.
Let ≤ r < ∞ . Then ˜ E pqr +1 ∼ = 0 for p ≤ − , and there areexact sequences → I p G p + q ( A ) I p +1 G p + q ( A ) → ˜ E pqr +1 → I r G p + q +1 ( N p +1 ) ( ι p +1 p ) ∗ −−−−−→ I r +1 G p + q +1 ( N p ) → for ≤ p ≤ r and → I r G p + q ( N p − r ) I r +1 G p + q ( N p − r ) → ˜ E pqr +1 → I r G p + q +1 ( N p +1 ) ( ι p +1 p ) ∗ −−−−−→ I r +1 G p + q +1 ( N p ) → for p ≥ r . For r = ∞ , let g Bad pq := \ r ∈ N I r G q ( N p ) , OMOLOGICAL ALGEBRA IN BIVARIANT K-THEORY 21 then we get E ∞ pq ∼ = 0 for p ≤ − , and exact sequences → I p G p + q ( A ) I p +1 G p + q ( A ) → ˜ E pq ∞ → g Bad p +1 ,p + q +1 → g Bad p,p + q +1 . Therefore, if T r ∈ N I r G q ( N p ) = 0 for all p, q , then ˜ E pq ∞ ∼ = I p G p + q ( A ) I p +1 G p + q ( A ) , that is, the ABC spectral sequence converges towards G m ( A ) , and the induced de-creasing filtration on the limit is (cid:0) I r G m ( A ) (cid:1) r ∈ N .Proof. This is proved exactly as in the homological case. (cid:3)
Notice that we do not claim that the maps g Bad p +1 ,p + q +1 → g Bad p,p + q +1 aresurjective. If G is representable, that is, G ( A ) ∼ = T ( A, B ) for some B ∈∈ T , thenthe question whether the maps g Bad p +1 ,p + q +1 → g Bad p,p + q +1 are surjective is relatedto the question whether I ∞ · I = I ∞ . In this case, g Bad p, ∗ = I ∞ ( N p [ ∗ ] , B ) . Sincethe maps ι n +1 n are I -versal, range (cid:0) g Bad p +1 , ∗ → g Bad p, ∗ (cid:1) = I ∞ ◦ I ( N p [ ∗ ] , B ) ⊆ I ∞ ( N p [ ∗ ] , B ) . Theorem 4.8.
Starting with the second page, the ABC spectral sequences for ho-mological and cohomological functors are independent of auxiliary choices and func-torial in A . Their second pages contain the derived functors: E pq ∼ = L p F q ( A ) , ˜ E pq ∼ = R p G q ( A ) . Proof.
We only formulate the proof for homological functors; the cohomologicalcase is similar. The map d := jk : E → E is induced by the maps δ p : P p +1 → P p [1] in the phantom tower. By Lemma 3.3, these maps form a P -projective resolutionof A . This together with counting of suspensions yields the description of E pq .Let f : A → A ′ be a morphism in T . By Lemma 3.3, it lifts to a morphismbetween the phantom towers over A and A ′ . This induces a morphism of exactcouples and hence a morphism of spectral sequences. The maps P n → P ′ n form achain map between the I -projective resolutions embedded in the phantom towers.This chain map lifting of f is unique up to chain homotopy (see [18, Proposition3.36]). Hence the induced map on E is unique and functorial. We get E r for r ∈ N ≥ ∪ {∞} as subquotients of E . Since our map on E is part of a morphismof exact couples, it descends to these subquotients in a unique and functorial way.Thus E r is functorial for all r ≥ . (cid:3) The naturality of the ABC spectral sequence does not mean that the exactsequences in Propositions 4.3 and 4.7 are natural. They use the exact couple under-lying the ABC spectral sequence, and this exact couple is not natural. It is easy tocheck that the maps E r +1 pq → F p + q : I p +1 ( A ) (cid:14) F p + q : I p ( A ) in Proposition 4.3 arecanonical for ≤ p ≤ r ≤ ∞ . But F p + q : I r +1 ( N p − r ) (cid:14) F p + q : I r ( N p − r ) dependson auxiliary choices.Our results so far only formulate the convergence problem for the ABC spectralsequence. It remains to check whether the relevant obstructions vanish. The easyspecial case where the projective resolutions have finite length is already dealt within [7]. Recall that P n I denotes the class of I n -projective objects. Lemma 4.9.
Let k ∈ N and A ∈∈ P k I . Then ι m + km = 0 and N m ∈∈ P k I for all m ∈ N .Proof. Since ι m + km is I k -versal, we have ι m + km = 0 if and only if N m ∈∈ P k I . Weprove ι m + km = 0 by induction on m . The case m = 0 is clear because N = A . If ι m + km = 0 , then ι m + km +1 factors through the map ǫ m : N m +1 → P m [1] by the longexact homology sequence for the triangles (4.1). If we compose the resulting map P m [1] → N m + k with ι m + k +1 m + k ∈ I , we get zero because P m [1] ∈∈ P . Thus ι m + k +1 m +1 vanishes as well. (cid:3) Proposition 4.10.
Let F : T → Ab be a homological functor and let m ∈ N .If A ∈∈ P m +1 I , then the ABC spectral sequence for F and A collapses at E m +2 and converges towards F ∗ ( A ) , and its limiting page E ∞ = E m +2 is supported inthe region ≤ p ≤ m .If, instead, A has a P -projective resolution of length m , then the ABC spectralsequence for F and A is supported in the region ≤ p ≤ m from the second pageonward, so that it collapses at E m +1 . If, in addition, A belongs to the localisingsubcategory of T generated by P I , then the spectral sequence converges towards F ∗ ( A ) .Similar assertions hold in the cohomological case.Proof. We only write down the argument in the homological case. If A ∈∈ P m +1 I ,then N p ∈∈ P m +1 I for all p ∈ N by Lemma 4.9. Therefore, F : I r ( N p ) = F ( N p ) forall r ≥ m + 1 . Plugging this into Proposition 4.3, we get E rpq = 0 for r ≥ m + 2 and p ≥ m + 1 . This forces the boundary maps d r to vanish for r ≥ m + 2 , so that E ∞ pq = E m +2 pq .Suppose now that A has a projective resolution of length m . Embed such aresolution in a phantom tower by Lemma 3.3, so that P p = 0 and N p ∼ = N p +1 for p > m . Then E r is supported in the region ≤ p ≤ m for all r ≥ . For r ≥ , thisholds for any choice of phantom tower by Theorem 4.8. As a consequence, d r = 0 for r ≥ m + 1 and hence E ∞ = E m +1 .Suppose, in addition, that A belongs to the localising subcategory of T generatedby P I . We claim that N p ∼ = 0 for p > m . This implies that the ABC spectralsequence converges towards F ∗ ( A ) . If D ∈ P I , then there are exact sequences T ∗ +1 ( D, N p ) T ∗ ( D, P p − ) ։ T ∗ ( D, N p − ) for all p ∈ N ≥ (see the proof of Lemma 3.3). Therefore, T ∗ ( D, N p ) = 0 for p > m if D ∈∈ P I . The class of objects D with this property is localising, that is, closedunder suspensions, direct sums, direct summands, and exact triangles. Hence itcontains the localising subcategory generated by P I . This includes A = N byassumption. Since it contains all P n as well, it contains N n for all n ∈ N because ofthe exact triangles in the phantom tower, so that T ∗ ( N n , N p ) vanishes for all n ∈ N .Thus T ∗ ( N p , N p ) = 0 , forcing N p = 0 . (cid:3) If A belongs to the localising subcategory generated by the I -projective ob-jects, then the existence of a projective resolution of length n implies that A is I n -projective. This fails without an additional hypothesis because I -contractibleobjects have projective resolutions of length .The converse assertion is usually far from true (see [7]). Proposition 2.2 showsthat an object A of T is I -projective if and only if there is an exact triangle P → P → A → P [1] with I -projective objects P and P . The resulting chaincomplex → P → P → A is an I -projective resolution if and only if the map A → P [1] is an I -phantom map, if and only if the map P → P is I -monic. Butthis need not be the case in general.Recall that the derived functors of the contravariant functor A T ∗ ( A, B ) are the extension groups Ext ∗ T , I ( A, B ) . These agree with extension groups in theAbelian approximation, that is, the target category of the universal I -exact stablehomological functor. In particular, Ext T , I ( A, B ) is the space of morphisms between OMOLOGICAL ALGEBRA IN BIVARIANT K-THEORY 23 the images of A and B in the Abelian approximation (see [18]). Theorem 4.8 andProposition 4.7 yield exact sequences(4.11) → T ( A, B ) I ( A, B ) → Ext T , I ( A, B ) → I ( N [ − , B ) ( ι ) ∗ −−−→ I ( A [ − , B ) → and(4.12) → I ( A, B ) I ( A, B ) → Ext T , I ( A, B ) → I ( N [ − , B ) ( ι ) ∗ −−−→ I ( N [ − , B ) → . In particular, we get injective maps T / I ( A, B ) → Ext T , I ( A, B ) , I / I ( A, B ) → Ext T , I ( A, B ) . But these maps need not be surjective. What they do is easy to understand. Thefirst map is simply the functor from T to the Abelian approximation. The secondmap embeds a morphism in I in an exact triangle. This triangle is I -exact be-cause it involves a phantom map, and hence provides an extension in the Abelianapproximation.The higher quotients I n / I n +1 ( A, B ) are also related to Ext n T , I ( A, B ) , but thisis merely a relation in the formal sense, that is, it is no longer a map. To constructthis relation, we use the I n -versality of the map ι n : A → N n . Thus any map f ∈ I n ( A, B ) factors through a map ˆ f : N n → B , and we can choose ˆ f ∈ I ( N n , B ) if f ∈ I n +1 ( A, B ) . Now we use the map T / I ( N n , B ) → Ext n T , I ( A, B ) provided by Theorem 4.8 and Proposition 4.7. Since f does not determine the classof ˆ f in T / I ( N n , B ) uniquely, we only get a relation. The ambiguity in this relationdisappears on the n th page of the ABC spectral sequence by Proposition 4.7.Once we have chosen ˆ f as above, we can also extend it to a morphism betweenthe phantom towers of A and B that shifts degrees down by n – the extension to theleft of N n is induced by ˆ f ◦ ι nm : N m → B for m < n and vanishes on P m for m < n .Thus we get a morphism between the ABC spectral sequences for A and B for anyhomological or cohomological functor – shifting degrees down by n , of course.4.2. An equivalent exact couple.
The cellular approximation tower produces aspectral sequence in the same way as the phantom tower.We extend the phantom tower to n < by ˜ A n = 0 and P n = 0 for all n < .The triangles (3.9) are exact for all n ∈ Z . A homological functor F maps theseexact triangles to exact chain complexes · · · / / F m ( ˜ A n ) α n +1 n ∗ / / F m ( ˜ A n +1 ) σ n ∗ / / F m ( P n ) κ n ∗ / / F m − ( ˜ A n ) / / · · · . As above, these amount to an exact couple D ′ i ′ / / D ′ j ′ (cid:3) (cid:3) (cid:8)(cid:8)(cid:8)(cid:8)(cid:8)(cid:8) E ′ k ′ [ [ D ′ pq := F p + q ( ˜ A p +1 ) ,E ′ pq := F p + q ( P p ) , i ′ pq := ( α p +2 p +1 ) ∗ : D ′ p,q → D ′ p +1 ,q − ,j ′ pq := ( σ p ) ∗ : D ′ p,q → E ′ p,q ,k ′ pq := ( κ p ) ∗ : E ′ p,q → D ′ p − ,q . Part of the commuting diagram (3.10) asserts that the identity maps E → E ′ and the maps γ p +1 , ∗ : D pq → D ′ pq form a morphism of exact couples between theexact couples from the phantom tower and the cellular approximation tower. Thisinduces a morphism between the resulting spectral sequences. Since this morphismacts identically on E , the induced morphisms on E r must be invertible for all r ∈ N ≥ ∪{∞} . Hence our new spectral sequence is isomorphic to the ABC spectralsequence.Although the spectral sequences are isomorphic, the underlying exact couplesare different and thus provide isomorphic but different descriptions of E ∞ .An important difference between the two exact couples is that D ′ pq = 0 for p ≤ .Hence any element of D ′ pq is annihilated by a sufficiently high power of i . Therefore,the kernel in (4.5) vanishes and E ∞ pq ∼ = D ′ p − ,q ∩ ker i ′ ∩ \ r ∈ N ( i ′ ) r ( D ′ ) . Let L pq := lim −→ range (cid:0) α rp ∗ : F q ( ˜ A p ) → F q ( ˜ A r ) (cid:1) ; these spaces define an increasing filtration ( F pq ) p ∈ N on lim −→ F q ( ˜ A r ) – we form thelimit with the maps α nm . Using (4.5) and the exactness of colimits of Abelian groups,we get isomorphisms E ∞ pq ∼ = L p +1 ,p + q L p,p + q . Hence E ∞ pq converges towards lim −→ F q ( ˜ A r ) and induces the filtration ( L p,p + q ) on itslimit – without any assumption on the ideal or the homological functor.In the cohomological case, the exact triangles (3.9) yield an exact couple aswell, and the morphism between the two exact couples from the cellular approxima-tion tower and the phantom tower induces an isomorphism between the associatedspectral sequences. Again, we get a new description of ˜ E ∞ .But the result is not as simple as in the homological case because the projectivelimit functor for Abelian groups is not exact. Let ˜ L pq := \ r ≥ p range (cid:0) α r ∗ p : G q ( ˜ A r ) → G q ( ˜ A p ) (cid:1) . Then ˜ E pq ∞ ∼ = ker (cid:0) ˜ L p +1 ,p + q → ˜ L p,p + q (cid:1) . In general, we cannot say much more than this. If ˜ L pq is the range of the map lim ←− r G q ( ˜ A r ) → G q ( ˜ A p ) for all p and q , then the ABC spectral sequence convergestowards lim ←− r G q ( ˜ A r ) and induces on this limit the decreasing filtration by the sub-spaces ker (cid:0) lim ←− r G q ( ˜ A r ) → G q ( ˜ A p ) (cid:1) for p ∈ N .5. Convergence of the ABC spectral sequence
There is an obvious obstruction to the convergence of the ABC spectral sequence:the subcategory N I of I -contractible objects. Since I -derived functors vanish on N I ,the spectral sequence cannot converge towards the original functor unless it vanisheson N I as well. At best, the ABC spectral sequence may converge to the localisation of the given functor at N I . We show that this is indeed the case for homologicalfunctors that commute with direct sums, provided the ideal I is compatible withdirect sums. The situation for cohomological functors is less satisfactory becausethe projective limit functor for Abelian groups is not exact.We continue to assume throughout this section that the category T has countabledirect sums. Various notions of convergence of spectral sequences are discussedin [4]. The following results deal only with strong convergence. Theorem 5.1.
Let I be a homological ideal compatible with direct sums in a tri-angulated category T ; let F : T → Ab be a homological functor that commutes with OMOLOGICAL ALGEBRA IN BIVARIANT K-THEORY 25 countable direct sums, and let A ∈∈ T ; let L F be the localisation of F at N I . Then F : I ∞ ( A ) = F : I ∞ ( A ) = [ r ∈ N F : I r ( A ) = range (cid:0) L F ( A ) → F ( A ) (cid:1) , and the ABC spectral sequence for F and A converges towards L F ( A ) with thefiltration (cid:0) L F : I k ( A ) (cid:1) k ∈ N . We have S k ∈ N L F : I k ( A ) = L F ( A ) .Proof. Lemma 3.7 implies that F : I r ( A ) is the range of α r ∗ : F ( ˜ A r ) → F ( A ) forall r ∈ N and that F : I ∞ ( A ) is the range of F (cid:0)L ˜ A r (cid:1) → F ( A ) because (3.29) isan I ∞ -projective resolution. Now F (cid:0)L ˜ A r (cid:1) = L F ( ˜ A r ) shows that F : I ∞ ( A ) isthe union of F : I r ( A ) for r ∈ N .Let ˜ A be as in (3.26), so that ˜ A = L ( A ) and L F ( A ) = F ( ˜ A ) . Since the inductivelimit functor for Abelian groups is exact, the map id − S on L F ( ˜ A r ) is injectiveand has cokernel lim −→ F ( ˜ A r ) . Since the top row in (3.26) is an exact triangle, thelong exact sequence yields F ( ˜ A ) ∼ = lim −→ F ( ˜ A r ) . As a consequence, the range of f ∗ : F ( ˜ A ) → F ( A ) is equal to F : I ∞ ( A ) . Since ˜ A is I ∞ -projective by Theorem 3.27and f : ˜ A → A is an I -equivalence, this map is an I ∞ -projective resolution of A .Hence the range of f ∗ also agrees with F : I ∞ ( A ) by Lemma 3.7.Especially, F : I ∞ ( A ) = F ( A ) if A ∈∈ h P I i . For such A , all objects that occur inthe phantom castle belong to h P I i as well, so that F ( N p ) = S r ∈ N F : I r ( N p ) for all p ∈ N . Hence Proposition 4.3 yields the convergence of the ABC spectral sequenceto F ( A ) as asserted. Since the ABC spectral sequences for A and ˜ A are isomorphicby Proposition 3.28, we get convergence towards L F ( A ) for general A . (cid:3) The convergence of the ABC spectral sequences is more problematic for a coho-mological functor G : T op → Ab because projective limits of Abelian groups are notexact. In the following, we assume that G maps direct sums to direct products –this is the correct compatibility with direct sums for contravariant functors.The exactness of the first row in (3.26) yields an exact sequence(5.2) lim ←− G ∗− ( ˜ A n ) G ∗ ( ˜ A ) ։ lim ←− G ∗ ( ˜ A n ) for any A (this also follows from [18, Theorem 4.4] applied to the ideal I ∞ ). Fur-thermore, G ∗ ( ˜ A ) = R G ∗ ( A ) . Since (3.26) is an I ∞ -projective resolution, we have lim ←− G − ( ˜ A n ) ∼ = I ∞ G ( ˜ A ) . The same argument as in the homological case yields(5.3) I ∞ G ( A ) = \ n ∈ N I n G ( A ) . Using compatibility of G with direct sums, we can also rewrite the obstructions tothe convergence of the ABC spectral sequence in Proposition 4.7: g Bad pq ∼ = I ∞ G q ( ˜ N p ) , where ˜ N p is the p th object in a phantom tower over ˜ A instead of A . The spectralsequence converges towards R G ( A ) if these obstructions all vanish. Proposition 5.4.
Let I be a homological ideal with enough projectives that is com-patible with direct sums, and let G : T op → Ab be a cohomological functor that mapsdirect sums to direct products. Let A ∈∈ T and let L ( A ) ∈∈ h P I i be its P I -cellularapproximation. If L ( A ) is I ∞ -projective, then the ABC spectral sequence for A and G converges towards R G ( A ) = G ◦ L ( A ) . Proof.
Proposition 3.28 implies that A and L ( A ) have canonically isomorphic ABCspectral sequences. Hence we may replace A by L ( A ) and assume that A itselfis I ∞ -projective. By Proposition 2.3, A is a direct summand of L n ∈ N A n with I n -projective objects A n . The ABC spectral sequence for each A n converges byProposition 4.10.Since I is compatible with countable direct sums, a direct sum of phantomcastles over A n is a phantom castle over L A n . Thus the ABC spectral sequencefor L n ∈ N A n is the direct product of the ABC spectral sequences for A n ; here weuse that G maps direct sums to direct products. Hence the ABC spectral sequencefor L n ∈ N A n converges towards Q G ( A n ) = G (cid:0)L A n (cid:1) . Since the ABC spectralsequence is an additive functor on T , this implies that the ABC spectral sequencefor any direct summand of L A n converges. This yields the convergence of theABC spectral sequence for L ( A ) , as desired. (cid:3) A classical special case
Before we apply our results to equivariant bivariant K-theory, we briefly discussa more classical application in homological algebra, where we recover results byMarcel Bökstedt and Amnon Neeman [5] and where the ABC sectral sequencespecialises to a spectral sequence due to Alexander Grothendieck.Let A be an Abelian category with enough projective objects and exact countabledirect sums. Let Ho( A ) be the homotopy category of chain complexes over A .We require no finiteness conditions, so that Ho( A ) is a triangulated category withcountable direct sums. We are interested in the derived category of A and thereforewant to localise at the full subcategory N ⊆ Ho( A ) of exact chain complexes. Thissubcategory is localising because countable direct sums of exact chain complexesare again exact by assumption.The obvious functor defining this subcategory N is the functor H : Ho( A ) → A Z that maps a chain complex to its homology. The functor H is a stable homologicalfunctor that commutes with direct sums. Hence its kernel I H is a homological idealthat is compatible with direct sums.Let PA Z ⊆ A Z be the full subcategory of projective objects. Since we assume A to have enough projective objects, any object of A Z admits an epimorphism froman object in PA Z . It is easy to see that the left adjoint of the homology functor isdefined on PA Z and maps a sequence ( P n ) of projective objects to the chain complex ( P n ) with vanishing boundary map. Since this functor is clearly fully faithful, weuse it to view PA Z as a full subcategory of Ho( A ) , omitting the functor H ⊢ fromour notation. Using the criterion of [18], it is easy to check that the functor H above is the universal I H -exact homological functor.Theorems 3.31 and 3.32 apply here. The first one shows that ( h PA Z i , N ) isa complementary pair of subcategories. Thus h PA Z i is equivalent to the derivedcategory of A . Furthermore, any object of the derived category is a homotopycolimit of a diagram with entries in ( PA Z ) ⋆n for n ∈ N .Let F : A → Ab be an additive covariant functor that commutes with directsums. We extend F to an exact functor Ho( F ) : Ho( A ) → Ho( Ab ) . Let ¯ F q = H q ◦ Ho( F ) : Ho( A ) → Ab be the functor that maps a chain complex C • to the q th homology of Ho( F )( C • ) .This is a homological functor. Its derived functors with respect to I H are computedin [18]: for a chain complex C • , we have L p ¯ F q ( C • ) = ( L p F ) (cid:0) H q ( C • ) (cid:1) , OMOLOGICAL ALGEBRA IN BIVARIANT K-THEORY 27 that is, we apply the usual derived functors of F to the homology of C • . Thus theABC spectral sequence computes the homology of the total derived functor of F applied to C • in terms of the derived functors of F , applied to H ∗ ( C • ) . Such aspectral sequence was already constructed by Alexander Grothendieck.7. Construction of the Baum–Connes assembly map
Finally, we apply our general machinery to construct the Baum–Connes assemblymap with coefficients first for locally compact groups and then for certain discretequantum groups. In the group case, we get a simpler argument than in [16].7.1.
The assembly map for locally compact groups.
Let G be a second count-able locally compact group and let KK G be the G -equivariant Kasparov category;its objects are the separable C ∗ -algebras with a strongly continuous action of G ,its morphism space A → B is KK G ( A, B ) . It is shown in [16] that this category istriangulated (we must exclude Z / -graded C ∗ -algebras for this).The category KK G has countable direct sums – they are just direct sums ofC ∗ -algebras. But uncountable direct sums usually do not exist because of the sepa-rability assumption in the definition of KK G , which is needed to make the analysiswork. Alternative definitions of bivariant K-theory by Joachim Cuntz [8] still workfor non-separable C ∗ -algebras, but it is not clear whether direct sums of C ∗ -algebrasremain direct sums in this category because the definition of the Kasparov groupsfor inseparable C ∗ -algebras involves colimits, which do not commute with the directproducts that appear in the definition of the direct sum.With enough effort, it should be possible to extend KK G to a category withuncountable direct sums. But it seems easier to avoid this by imposing cardinalityrestrictions on direct sums.For any closed subgroup H ⊆ G , we have induction and restriction functors Ind GH : KK H → KK G , Res HG : KK G → KK H ; the latter functor is quite trivial and simply forgets part of the group action. Thesefunctors give rise to two subcategories of KK G , which play a crucial role in [16]. Definition 7.1.
Let F be the set of all compact subgroups of G . CC := { A ∈∈ KK G | Res HG ( A ) = 0 for all H ∈ F} , CI := { Ind GH ( A ) | A ∈∈ KK H and H ∈ F} . Whereas the subcategory CC is localising by definition, CI is not. Therefore, thelocalising subcategory it generates, hCIi , plays an important role as well. Since G acts properly on objects of CI , they satisfy the Baum–Connes conjecture, that is,the Baum–Connes assembly map is an isomorphism for coefficients in CI . Sincedomain and target of the assembly map are exact functors on KK G , this extends tothe category hCIi . On objects of CC the domain of the Baum–Connes assembly mapis known to vanish, so that the Baum–Connes conjecture predicts K ∗ ( G ⋉ r A ) = 0 for A ∈∈ CC .On a technical level, the main tool in [16] is that the pair of subcategories ( hCIi , CC ) is complementary. Hence the Baum–Connes assembly map is determinedby what it does on these two subcategories. This implies that its domain is thelocalisation of the functor A K ∗ ( G ⋉ r A ) at CC and that the assembly map isthe natural transformation from this localisation to the original functor.Put differently, the Baum–Connes assembly map is the only natural transforma-tion from an exact functor on KK G to the functor K ∗ ( G ⋉ r ) that is an isomorphismon CI and whose domain vanishes on CC (we give some more details about this ar-gument in the related quantum group case below). In order to prove that ( hCIi , CC ) is complementary, we introduce the followingideal: Definition 7.2.
Let I = T H ∈F ker Res HG .This ideal consists of the morphisms that vanish for compact subgroups in thenotation of [16]. Clearly, an object belongs to CC if and only if its identity mapbelongs to I , that is, N I = CC . Moreover, [16, Proposition 4.4] implies that objectsof CI are I -projective; even more, f ∈ I ( A, B ) if and only if f induces the zeromap KK G ∗ ( D, A ) → KK G ∗ ( D, B ) for all D ∈∈ CI .We can also describe I as the kernel of a single functor: F = (Res HG ) H ∈F : KK G → Y H ∈F KK H . The functor F commutes with direct sums because each functor Res HG clearly doesso. Hence I is compatible with countable direct sums.The following theorem contains the main assertion in [16, Theorem 4.7]. We willprovide a simpler proof here than in [16]. Theorem 7.3.
The projective objects for I are the retracts of direct sums of objectsin CI , and the ideal I has enough projective objects. Hence the pair of subcategories ( hCIi , CC ) is complementary.Proof. As in Theorem 3.31, we study the partially defined left adjoint of the func-tor F above or, equivalently, of the functors Res HG for H ∈ F .The discrete case is particularly simple because then all H ∈ F are open sub-groups. If H ⊆ G is open, then Ind GH is left adjoint to Res HG . Thus we may take PC = Q H ∈F KK H in Theorem 3.31 and get F ⊢ (cid:0) ( A H ) H ∈F (cid:1) = L H ∈F Ind GH ( A H ) .Notice that the set F is countable if G is discrete, so that this definition is legiti-mate. It follows that I has enough projective objects and that they are all directsummands of L H ∈F Ind GH ( A H ) for suitable families ( A H ) , as asserted.For locally compact G , the argument gets more complicated because the functor Res HG does not always have a left adjoint, and if it has, it need not be simply Ind GH . But there are still enough compact subgroups H for which the left adjointis defined on enough H -C ∗ -algebras and close enough to the induction functor forthe argument above to go through.A good way to understand this is the duality theory developed in [9, 10]. Thisis relevant because the induction functor provides an equivalence of categories KK H ≃ KK G ⋉ G/H , where we use the groupoid G ⋉ G/H , that is, we consider G -equivariant bundles of C ∗ -algebras over G/H . This equivalence of categoriesreflects the equivalence between the groupoids H and G ⋉ G/H .Identifying KK H ≃ KK G ⋉ G/H , the restriction functor
Res HG becomes the functor p ∗ G/H : KK G → KK G ⋉ G/H that pulls back a G -C ∗ -algebra to a trivial bundle of G -C ∗ -algebras over G/H . Following [12], it is shown in [9] that the left adjoint of p ∗ G/H is defined on all trivial bundles if
G/H is a smooth manifold. We will seethat this is enough for our purposes.As in [16], we call a compact subgroup large if it is a maximal compact subgroupin an open, almost connected subgroup of G .Let H be large. Then G/H is a smooth manifold and any compact subgroupis contained in a large one by [16, Lemma 3.1]. Furthermore, since G is secondcountable there is a sequence ( U n ) n ∈ N of almost connected open subgroups of G such that any other one is contained in U n for some n ∈ N . Pick a maximalcompact subgroup H n ⊆ U n for each n ∈ N . Then any compact subgroup of G is OMOLOGICAL ALGEBRA IN BIVARIANT K-THEORY 29 subconjugate to H n for some n ∈ N . Therefore, we already have I = T n ∈ N Res H n G because Res KG factors through Res HG if K is subconjugate to H .For a compact subgroup H ⊆ G , let RKK G ( G/H ) ⊆ KK G ⋉ G/H be the fullsubcategory of trivial bundles over
G/H or, equivalently, the essential range of thefunctor p ∗ G/H . We do not care whether this category is triangulated, it is certainlyadditive. We replace the functors
Res HG by p ∗ G/H : KK G → RKK G ( G/H ) for H ∈ F .For the large compact subgroups H n selected above, the results in [10] show thatthe left adjoint of p ∗ G/H is defined on all of
RKK G ( G/H ) and maps the trivialbundle with fibre A to C ( T G/H ) ⊗ A with the diagonal action of G ; here T G/H denotes the tangent space of
G/H (we are not allowed to use the Clifford algebradual considered in [9] because it involves Z / -graded C ∗ -algebras, which do notbelong to our category).Thus we have verified the hypotheses of Theorem 3.31 and can conclude that I has enough projective objects and that CC is reflective. It remains to observethat the projective objects are precisely the direct summands of countable di-rect sums of objects of CI . We have already observed that objects of CI are I -projective. Conversely, Theorem 3.31 shows that the projective objects are re-tracts of L n ∈ N C ( T G/H n ) ⊗ A n for suitable G -C ∗ -algebras A n . The summandsare isomorphic to Ind GH n ( C ( T G/H n ) ⊗ A n ) where T G/H n denotes the tangentspace of G/H n at the base point · H n . Hence all projective objects are of therequired form. (cid:3) Since the stable homological functor F ∗ ( A ) := K ∗ ( G ⋉ r A ) commutes with directsums, Theorem 5.1 applies to it and shows that the ABC spectral sequence for theideal I converges towards the domain of the Baum–Connes assembly map – whichis the localisation of F ∗ at CC by [16, Theorem 5.2].It turns out that for a totally disconnected group G the ABC spectral sequenceagrees with a known spectral sequence that we get from the older definition of theBaum–Connes assembly map and the skeletal filtration of a G -CW-model for theuniversal proper G -space E G (see [13]). We omit the proof of this statement, whichrequires some work.7.2. An assembly map for torsion-free discrete quantum groups.
Beforewe turn to the assembly map, we must discuss some open problems that lead us torestrict attention to the torsion-free case.The first issue is the correct definition of “torsion” for locally compact quantumgroups. The torsion in a locally compact group is the family of compact sub-groups. Quantum groups exhibit some torsion phenomena that do not appear forgroups, and it is conceivable that we have not yet found all of them. First, compactquantum subgroups are not enough: they should be replaced by proper quantumhomogeneous spaces, so that open subgroups provide torsion in C ∗ ( G ) whenever G is disconnected. Secondly, projective representations of compact groups with a non-trivial cocycle also provide torsion (in their discrete dual); for instance, C ∗ (cid:0) SO (3) (cid:1) is not torsion-free because of its projective representation on C .If we considered C ∗ (cid:0) SO (3) (cid:1) to be torsion-free, then the Baum–Connes assemblymap for it (which we describe below) would fail to be an isomorphism. The correctformulation of the Baum–Connes conjecture for C ∗ (cid:0) SO (3) (cid:1) turns out to be equiv-alent to the Baum–Connes conjecture for C ∗ (cid:0) SU (2) (cid:1) – which is torsion-free – sothat there is no need to discuss it in its own right in [17].I propose to approach torsion in discrete quantum groups by studying actionsof its compact dual quantum group on finite-dimensional C ∗ -algebras. A discrete quantum group is torsion-free if any such action is a direct sum of actions that areMorita equivalent to the trivial action on C .The above definition of torsion gives the expected results in simple cases. First,C ( G ) for a discrete group G is torsion-free if and only if G contains no finitesubgroups. Secondly, C ∗ ( G ) for a compact group is torsion-free if and only if G isconnected and has torsion-free fundamental group; this is exactly the generality inwhich Universal Coefficient Theorems for equivariant Kasparov theory work (see[17, 22]). Christian Voigt shows in [24] that the quantum deformations of simplyconnected Lie groups such as SU q ( n ) are torsion-free.Another issue is to find analogues of the restriction and induction functors for thenon-classical torsion that may appear, and to prove analogues of the adjointnessrelations used in the proof of Theorem 7.3. For honest quantum subgroups, therestriction functor is evident, and Stefaan Vaes has constructed induction functorsfor actions of quantum group C ∗ -algebras in [23]. I expect restriction to be leftadjoint to induction for open quantum subgroups and, in particular, for quantumsubgroups of discrete quantum groups.For the time being, we avoid these problems and limit our attention to thetorsion-free case. More precisely, we consider arbitrary discrete quantum groups,but disregard torsion. The resulting assembly map should not be an isomorphismfor quantum groups with torsion.The discrete quantum groups are precisely the duals of compact quantum groups;we use reduced duals here because these appear also in the Baum–Connes conjecture.It is useful to reformulate results about a discrete quantum group in terms of itscompact dual as in [18, Remark 2.9]. Let G be a compact quantum group andlet b G be its discrete dual. Since we pretend that b G is torsion-free, there is only one“restriction functor” to consider: the forgetful functor KK b G → KK that forgets theaction of b G altogether. The category KK b G is equivalent to KK G by Baaj–Skandalisduality. Under this equivalence, the forgetful functor KK b G → KK corresponds tothe crossed product functor G ⋉ : KK G → KK , A G ⋉ A. The induction functor from the trivial subgroup to b G corresponds under Baaj–Skandalis duality to the functor τ : KK → KK G that equips a C ∗ -algebra with thetrivial action of G . This functor is left adjoint to the crossed product functor.Hence the relevant subcategories CI , CC and the ideal I correspond to CI = { τ ( A ) | A ∈∈ KK } , CC = { A ∈∈ KK G | G ⋉ A ≃ } , where ≃ means KK-equivalence, that is, isomorphism in KK , and I = { f ∈ KK G | G ⋉ f = 0 } . The ideal I is already studied in [18, §5]. It is shown there that I has enoughprojective objects, and the universal homological functor for it is described. Thetarget category involves actions of the representation ring Rep( G ) of the compactquantum group G on objects of KK ; such an action on A is, by definition, a ring ho-momorphism Rep( G ) → KK ( A, A ) . The category KK [Rep( G )] of Rep( G ) -modulesin KK is not yet Abelian because KK is not Abelian. To remedy this, we must re-place KK by its Freyd category of coherent functors KK → Ab . But this completiondoes not affect homological algebra much because KK [Rep( G )] is an exact subcat-egory that contains all projective objects; hence we can compute derived functorswithout leaving the subcategory KK [Rep( G )] .We could modify the ideal I and consider all f for which G ⋉ f induces thezero map on K-theory. This leads to a simpler Abelian approximation, namely, OMOLOGICAL ALGEBRA IN BIVARIANT K-THEORY 31 the category of all countable Z / -graded Rep( G ) -modules. But this larger ideal nolonger leads to the subcategories CC and CI above. Theorem 7.4.
Let G be any compact quantum group. Then the ideal I is com-patible with countable direct sums and has enough projective objects. The pair ofsubcategories ( hCIi , CC ) is complementary.Proof. The ideal I has enough I -projective objects by [16, Lemma 5.2], which alsoshows that the I -projective objects are precisely the direct summands of objectsin CI . The ideal I is compatible with direct sums because the crossed productfunctor commutes with direct sums. Now Theorem 3.21 shows that the pair ofsubcategories ( hCIi , CC ) is complementary. (cid:3) Definition 7.5.
Let F : KK G → A be some homological functor. The assemblymap for F with coefficients in A is the map L F ( A ) → F ( A ) , where the localisation L F is formed with respect to the subcategory CC .To get an analogue of the Baum–Connes assembly map, we should considerthe functor F ( A ) := K ∗ ( A ) because it corresponds to the functor B K ∗ ( G ⋉ r B ) under Baaj–Skandalis duality. A torsion-free discrete quantum group has the Baum–Connes property with coefficients if the assembly map L F ( A ) → F ( A ) is anisomorphism for all A for this functor. Proposition 7.6.
Let F : KK G → A be a homological functor that commutes withdirect sums. The assembly map L F ⇒ F is the unique natural transformation froma functor ˜ F to F with the following properties: • ˜ F is homological and commutes with direct sums; • the natural transformation is an isomorphism for objects in CI ; • ˜ F vanishes on CC .Proof. Let ˜ F ⇒ F be a natural transformation with the required properties. Sinceboth functors involved are homological, the Five Lemma implies that the class ofobjects for which the natural transformation ˜ F ⇒ F is an isomorphism is trian-gulated. It is also closed under direct sums because both functors commute withdirect sums. Hence the natural transformation ˜ F ⇒ F is an isomorphism for allobjects in hCIi because this holds for objects in CI .Since ˜ F vanishes on CC and is homological, the universal property of the local-isation shows that the natural transformation ˜ F ⇒ F factors uniquely throughthe assembly map: ˜ F ⇒ L F ⇒ F . Both ˜ F and L F descend to the category KK G / CC , which is equivalent to hCIi . Since both natural transformations ˜ F ⇒ F and L F ⇒ F are invertible on objects of hCIi , we get the desired natural isomor-phism ˜ F ∼ = L F . (cid:3) The critical property in Proposition 7.6 is the vanishing on CC . This cannot beexpected if b G has torsion. The Baum–Connes assembly map is an isomorphismfor F if and only if F vanishes on CC : one direction is trivial, and the other followsby taking ˜ F = F in Proposition 7.6. While this reformulation of the Baum–Connesconjecture came too late to be used in verifying the conjecture for groups, it isquite helpful for duals of compact groups (see [17]) and probably also for theirdeformations. 8. Conclusion
The idea of localisation – central both in homological algebra and in homotopytheory – is becoming more important in non-commutative topology as well. Whenrefined using homological ideals, it unifies various new and old universal coefficienttheorems, the Baum–Connes conjecture, and its extensions to quantum groups.
Homological ideals provide some basic topological tools in the general setting oftriangulated categories. This includes • important notions from homological algebra like projective resolutions andderived functors (these were already dealt with in [18]); • an efficient method to check that pairs of subcategories in a triangulatedcategory are complementary; • some control on how objects of the category are constructed from generators,that is, from the projective objects for the ideal; • a natural spectral sequence that computes the localisation of a homologicalfunctor from its values on generators.Since the assumptions on the underlying category are quite weak, all this appliesto equivariant bivariant K-theory.We have applied this general machinery to construct the Baum–Connes assemblymap for torsion-free quantum groups, whose domain is of topological nature in thesense that it can be computed by topological techniques such as spectral sequences.But much remains to be done here. The three main issues are to understandtorsion in locally compact quantum groups, to adapt the reduction and inductionfunctors to exotic torsion phenomena, and to check whether the assembly map isan isomorphism. These problems are mainly analytical in nature. References [1] J. F. Adams,
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